TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS

HOUS = α + β 1 DPI + β 2 PRELHOUS + u

In this sequence we will make an initial exploration of the determinants of aggregate consumer expenditure on housing services using the Demand Functions data set. 1

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS

HOUS = α + β 1 DPI + β 2 PRELHOUS + u

HOUS is aggregate consumer expenditure on housing services and DPI is aggregate disposable personal income. Both are measured in $ billions at 1992 constant prices. 2

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS

PHOUS PRELHOUS = 100 × PTPE

HOUS = α + β 1 DPI + β 2 PRELHOUS + u

PRELHOUS is a relative price index for housing services constructed by dividing the nominal price index for housing services by the price index for total personal expenditure. 3

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS 104

PHOUS PRELHOUS = 100 × PTPE

102 100 98 96 94 92 90 88 86 1959

1963

1967

1971

1975

1979

1983

1987

1991

Here is a plot of PRELHOUS for the sample period, 1959-1994.

4

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: HOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C 2.654701 28.91571 0.091808 0.9274 DPI 0.151521 0.001243 121.9343 0.0000 PRELHOUS -0.556949 0.290640 -1.916285 0.0640 ============================================================== R-squared 0.997811 Mean dependent var 429.3306 Adjusted R-squared 0.997679 S.D. dependent var 149.1037 S.E. of regression 7.183749 Akaike info criter 4.023298 Sum squared resid 1703.006 Schwarz criterion 4.155258 Log likelihood -120.5012 F-statistic 7522.482 Durbin-Watson stat 0.809993 Prob(F-statistic) 0.000000 ============================================================== Here is the regression output using EViews. It was obtained by loading the workfile, clicking on Quick, then on Estimate, and then typing HOUS C DPI PRELHOUS in the box. Note that in EViews you must include C in the command if your model has an intercept. 5

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: HOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C 2.654701 28.91571 0.091808 0.9274 DPI 0.151521 0.001243 121.9343 0.0000 PRELHOUS -0.556949 0.290640 -1.916285 0.0640 ============================================================== R-squared 0.997811 Mean dependent var 429.3306 Adjusted R-squared 0.997679 S.D. dependent var 149.1037 S.E. of regression 7.183749 Akaike info criter 4.023298 Sum squared resid 1703.006 Schwarz criterion 4.155258 Log likelihood -120.5012 F-statistic 7522.482 Durbin-Watson stat 0.809993 Prob(F-statistic) 0.000000 ============================================================== Here is the regression output. We will start by interpreting the coefficients. The coefficient of DPI indicates that if aggregate income rises by $1 billion, aggregate expenditure on housing services rises by $151 millions. Is this a plausible figure? 6

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: HOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C 2.654701 28.91571 0.091808 0.9274 DPI 0.151521 0.001243 121.9343 0.0000 PRELHOUS -0.556949 0.290640 -1.916285 0.0640 ============================================================== R-squared 0.997811 Mean dependent var 429.3306 Adjusted R-squared 0.997679 S.D. dependent var 149.1037 S.E. of regression 7.183749 Akaike info criter 4.023298 Sum squared resid 1703.006 Schwarz criterion 4.155258 Log likelihood -120.5012 F-statistic 7522.482 Durbin-Watson stat 0.809993 Prob(F-statistic) 0.000000 ============================================================== Possibly. It implies that 15 cents out of the marginal dollar are spent on housing. Housing is the largest category of consumer expenditure, so we would expect a substantial coefficient. Perhaps it is a little low. 7

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: HOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C 2.654701 28.91571 0.091808 0.9274 DPI 0.151521 0.001243 121.9343 0.0000 PRELHOUS -0.556949 0.290640 -1.916285 0.0640 ============================================================== R-squared 0.997811 Mean dependent var 429.3306 Adjusted R-squared 0.997679 S.D. dependent var 149.1037 S.E. of regression 7.183749 Akaike info criter 4.023298 Sum squared resid 1703.006 Schwarz criterion 4.155258 Log likelihood -120.5012 F-statistic 7522.482 Durbin-Watson stat 0.809993 Prob(F-statistic) 0.000000 ============================================================== The coefficient of PRELHOUS indicates that a one-point increase in this price index causes expenditure on housing to fall by $557 millions. It is not easy to determine whether this is plausible. At least the effect is negative. 8

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: HOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C 2.654701 28.91571 0.091808 0.9274 DPI 0.151521 0.001243 121.9343 0.0000 PRELHOUS -0.556949 0.290640 -1.916285 0.0640 ============================================================== R-squared 0.997811 Mean dependent var 429.3306 Adjusted R-squared 0.997679 S.D. dependent var 149.1037 S.E. of regression 7.183749 Akaike info criter 4.023298 Sum squared resid 1703.006 Schwarz criterion 4.155258 Log likelihood -120.5012 F-statistic 7522.482 Durbin-Watson stat 0.809993 Prob(F-statistic) 0.000000 ============================================================== The constant has no meaningful interpretation. (Literally, it indicates that $2.655 billions would be spent on housing services if aggregate income and the price series were both 0.) 9

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: HOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C 2.654701 28.91571 0.091808 0.9274 DPI 0.151521 0.001243 121.9343 0.0000 PRELHOUS -0.556949 0.290640 -1.916285 0.0640 ============================================================== R-squared 0.997811 Mean dependent var 429.3306 Adjusted R-squared 0.997679 S.D. dependent var 149.1037 S.E. of regression 7.183749 Akaike info criter 4.023298 Sum squared resid 1703.006 Schwarz criterion 4.155258 Log likelihood -120.5012 F-statistic 7522.482 Durbin-Watson stat 0.809993 Prob(F-statistic) 0.000000 ============================================================== The explanatory power of the model appears to be excellent. The coefficient of DPI has a very high t statistic and R2 is close to a perfect fit. But the t statistic for the price series is a little disappointing. 10

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS

HOUS = αDPI β1 PRELHOUS β 2 v

Constant elasticity functions are usually considered preferable to linear functions in models of consumer expenditure. Here β1 is the income elasticity and β2 is the price elasticity for expenditure on housing services. 11

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS

HOUS = αDPI β1 PRELHOUS β 2 v

LGHOUS = log α + β 1 LGDPI + β 2 LGPRHOUS + log v

We linearize the model by taking logarithms. We will regress LGHOUS, the logarithm of expenditure on housing services, , on LGDPI, the logarithm of disposable personal income, and LGPRHOUS, the logarithm of the relative price index for housing services. 12

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -2.032685 0.322726 -6.298478 0.0000 LGDPI 1.132248 0.008705 130.0650 0.0000 LGPRHOUS -0.227634 0.065841 -3.457323 0.0015 ============================================================== R-squared 0.998154 Mean dependent var 5.996930 Adjusted R-squared 0.998042 S.D. dependent var 0.377702 S.E. of regression 0.016714 Akaike info criter -8.103399 Sum squared resid 0.009218 Schwarz criterion -7.971439 Log likelihood 97.77939 F-statistic 8920.496 Durbin-Watson stat 0.846451 Prob(F-statistic) 0.000000 ============================================================== Here is the regression output. The estimate of the income elasticity is 1.13. Is this plausible? 13

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -2.032685 0.322726 -6.298478 0.0000 LGDPI 1.132248 0.008705 130.0650 0.0000 LGPRHOUS -0.227634 0.065841 -3.457323 0.0015 ============================================================== R-squared 0.998154 Mean dependent var 5.996930 Adjusted R-squared 0.998042 S.D. dependent var 0.377702 S.E. of regression 0.016714 Akaike info criter -8.103399 Sum squared resid 0.009218 Schwarz criterion -7.971439 Log likelihood 97.77939 F-statistic 8920.496 Durbin-Watson stat 0.846451 Prob(F-statistic) 0.000000 ============================================================== Probably. Housing is an essential category of consumer expenditure, and necessities generally have elasticities lower than 1. But it also has a luxury component, in that people tend to move to more desirable housing as income increases. 14

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -2.032685 0.322726 -6.298478 0.0000 LGDPI 1.132248 0.008705 130.0650 0.0000 LGPRHOUS -0.227634 0.065841 -3.457323 0.0015 ============================================================== R-squared 0.998154 Mean dependent var 5.996930 Adjusted R-squared 0.998042 S.D. dependent var 0.377702 S.E. of regression 0.016714 Akaike info criter -8.103399 Sum squared resid 0.009218 Schwarz criterion -7.971439 Log likelihood 97.77939 F-statistic 8920.496 Durbin-Watson stat 0.846451 Prob(F-statistic) 0.000000 ============================================================== Thus an elasticity near 1 seems about right. The price elasticity is 0.23, suggesting that expenditure on this category is price-inelastic. 15

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -2.032685 0.322726 -6.298478 0.0000 LGDPI 1.132248 0.008705 130.0650 0.0000 LGPRHOUS -0.227634 0.065841 -3.457323 0.0015 ============================================================== R-squared 0.998154 Mean dependent var 5.996930 Adjusted R-squared 0.998042 S.D. dependent var 0.377702 S.E. of regression 0.016714 Akaike info criter -8.103399 Sum squared resid 0.009218 Schwarz criterion -7.971439 Log likelihood 97.77939 F-statistic 8920.496 Durbin-Watson stat 0.846451 Prob(F-statistic) 0.000000 ============================================================== Again, the constant has no meaningful interpretation.

16

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -2.032685 0.322726 -6.298478 0.0000 LGDPI 1.132248 0.008705 130.0650 0.0000 LGPRHOUS -0.227634 0.065841 -3.457323 0.0015 ============================================================== R-squared 0.998154 Mean dependent var 5.996930 Adjusted R-squared 0.998042 S.D. dependent var 0.377702 S.E. of regression 0.016714 Akaike info criter -8.103399 Sum squared resid 0.009218 Schwarz criterion -7.971439 Log likelihood 97.77939 F-statistic 8920.496 Durbin-Watson stat 0.846451 Prob(F-statistic) 0.000000 ============================================================== The explanatory power of the model appears to be excellent. Now even the price elasticity is significantly different from 0, suggesting that this may be a more appropriate specification. 17

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Current and Lagged Values of the Logarithm of Disposable Personal Income Year

LGDPI

LGDPI(-1)

LGDPI(-2)

1959 1960 1961 1962 1963 1964 ...... ...... 1990 1991 1992 1993 1994

5.2750 5.3259 5.3720 5.4267 5.4719 5.5175 ...... ...... 6.4413 6.4539 6.4720 6.4846 6.5046

5.2750 5.3259 5.3720 5.4267 5.4719 ...... ...... 6.4210 6.4413 6.4539 6.4720 6.4846

5.2750 5.3259 5.3720 5.4267 ...... ...... 6.3984 6.4210 6.4413 6.4539 6.4720

Next, we will introduce some simple dynamics. Expenditure on housing is subject to inertia and responds slowly to changes in income and price. Accordingly we will consider specifications of the model where it depends on lagged values of income and price. 18

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Current and Lagged Values of the Logarithm of Disposable Personal Income Year

LGDPI

LGDPI(-1)

LGDPI(-2)

1959 1960 1961 1962 1963 1964 ...... ...... 1990 1991 1992 1993 1994

5.2750 5.3259 5.3720 5.4267 5.4719 5.5175 ...... ...... 6.4413 6.4539 6.4720 6.4846 6.5046

5.2750 5.3259 5.3720 5.4267 5.4719 ...... ...... 6.4210 6.4413 6.4539 6.4720 6.4846

5.2750 5.3259 5.3720 5.4267 ...... ...... 6.3984 6.4210 6.4413 6.4539 6.4720

A variable X lagged one time period has values that are simply the previous values of X, and it is conventionally denoted X(-1). Here LGDPI(-1) has been derived from LGDPI. You can see, for example, that the value of LGDPI(-1) in 1994 is just the value of LGDPI in 1993. 19

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Current and Lagged Values of the Logarithm of Disposable Personal Income Year

LGDPI

LGDPI(-1)

LGDPI(-2)

1959 1960 1961 1962 1963 1964 ...... ...... 1990 1991 1992 1993 1994

5.2750 5.3259 5.3720 5.4267 5.4719 5.5175 ...... ...... 6.4413 6.4539 6.4720 6.4846 6.5046

5.2750 5.3259 5.3720 5.4267 5.4719 ...... ...... 6.4210 6.4413 6.4539 6.4720 6.4846

5.2750 5.3259 5.3720 5.4267 ...... ...... 6.3984 6.4210 6.4413 6.4539 6.4720

Similarly for the other years. Note that LGDPI(-1) is not defined for 1959, given the data set. Of course, in this case, we could obtain it from the 1960 issues of the Survey of Current Business. 20

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Current and Lagged Values of the Logarithm of Disposable Personal Income Year

LGDPI

LGDPI(-1)

LGDPI(-2)

1959 1960 1961 1962 1963 1964 ...... ...... 1990 1991 1992 1993 1994

5.2750 5.3259 5.3720 5.4267 5.4719 5.5175 ...... ...... 6.4413 6.4539 6.4720 6.4846 6.5046

5.2750 5.3259 5.3720 5.4267 5.4719 ...... ...... 6.4210 6.4413 6.4539 6.4720 6.4846

5.2750 5.3259 5.3720 5.4267 ...... ...... 6.3984 6.4210 6.4413 6.4539 6.4720

Similarly, LGDPI(-2) is LGDPI lagged 2 time periods. LGDPI(-2) in 1994 is the value of LGDPI in 1992, and so on. Generalizing, X(-s) is X lagged s time periods. 21

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 1994 Included observations: 35 after adjusting endpoints ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -1.856591 0.280604 -6.616409 0.0000 LGDPI(-1) 1.100832 0.007644 144.0081 0.0000 LGPRHOUS(-1) -0.203492 0.056687 -3.589766 0.0011 ============================================================== R-squared 0.998569 Mean dependent var 6.017555 Adjusted R-squared 0.998480 S.D. dependent var 0.362063 S.E. of regression 0.014117 Akaike info criter -5.601000 Sum squared resid 0.006378 Schwarz criterion -5.467684 Log likelihood 101.0175 F-statistic 11165.70 Durbin-Watson stat 1.121077 Prob(F-statistic) 0.000000 ============================================================== Here is a logarithmic regression of current expenditure on housing on lagged income and price. Note that EViews, in common with most regression applications, recognizes X(-1) as being the lagged value of X and there is no need to define it as a distinct variable.. 22

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS ============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 1994 Included observations: 35 after adjusting endpoints ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -1.856591 0.280604 -6.616409 0.0000 LGDPI(-1) 1.100832 0.007644 144.0081 0.0000 LGPRHOUS(-1) -0.203492 0.056687 -3.589766 0.0011 ============================================================== R-squared 0.998569 Mean dependent var 6.017555 Adjusted R-squared 0.998480 S.D. dependent var 0.362063 S.E. of regression 0.014117 Akaike info criter -5.601000 Sum squared resid 0.006378 Schwarz criterion -5.467684 Log likelihood 101.0175 F-statistic 11165.70 Durbin-Watson stat 1.121077 Prob(F-statistic) 0.000000 ============================================================== The estimate of the lagged income and price elasticities are 1.10 and 0.20, respectively.

23

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Alternative Dynamic Specifications, Housing Services Variable LGDPI

(1) 1.13 (0.01)

LGDPI(-1)

-

LGDPI(-2)

-

LGPRHOUS

-0.23 (0.07)

(2)

(3)

1.10 (0.01) -

LGPRHOUS(-1)

-

-0.20 (0.06)

LGPRHOUS(-2)

-

-

R2

0.998

0.999

(4)

(5)

-

0.38 (0.15)

0.33 (0.14)

-

0.73 (0.15)

0.28 (0.21)

1.07 (0.01)

-

0.48 (0.15)

-

-0.19 (0.08)

-0.13 (0.19)

-

0.14 (0.08)

0.25 (0.19)

-0.19 (0.06) 0.998

0.999

-0.33 (0.19) 0.999

The regression results will be summarized in a table for comparison. The results of the lagged-values regression are virtually identical to those of the current-values regression. 24

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Alternative Dynamic Specifications, Housing Services Variable LGDPI

(1) 1.13 (0.01)

LGDPI(-1)

-

LGDPI(-2)

-

LGPRHOUS

-0.23 (0.07)

(2)

(3)

1.10 (0.01) -

LGPRHOUS(-1)

-

-0.20 (0.06)

LGPRHOUS(-2)

-

-

R2

0.998

0.999

(4)

(5)

-

0.38 (0.15)

0.33 (0.14)

-

0.73 (0.15)

0.28 (0.21)

1.07 (0.01)

-

0.48 (0.15)

-

-0.19 (0.08)

-0.13 (0.19)

-

0.14 (0.08)

0.25 (0.19)

-0.19 (0.06) 0.998

0.999

-0.33 (0.19) 0.999

So also are the results of regressing LGHOUS on LGDPI and LGPRHOUS lagged two years.

25

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Alternative Dynamic Specifications, Housing Services Variable LGDPI

(1) 1.13 (0.01)

LGDPI(-1)

-

LGDPI(-2)

-

LGPRHOUS

-0.23 (0.07)

(2)

(3)

1.10 (0.01) -

LGPRHOUS(-1)

-

-0.20 (0.06)

LGPRHOUS(-2)

-

-

R2

0.998

0.999

(4)

(5)

-

0.38 (0.15)

0.33 (0.14)

-

0.73 (0.15)

0.28 (0.21)

1.07 (0.01)

-

0.48 (0.15)

-

-0.19 (0.08)

-0.13 (0.19)

-

0.14 (0.08)

0.25 (0.19)

-0.19 (0.06) 0.998

0.999

-0.33 (0.19) 0.999

One approach to discriminating between the effects of current and lagged income and price is to include both in the equation. Since both may be important, failure to do so may give rise to omitted variable bias. 26

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Alternative Dynamic Specifications, Housing Services Variable LGDPI

(1) 1.13 (0.01)

LGDPI(-1)

-

LGDPI(-2)

-

LGPRHOUS

-0.23 (0.07)

(2)

(3)

1.10 (0.01) -

LGPRHOUS(-1)

-

-0.20 (0.06)

LGPRHOUS(-2)

-

-

R2

0.998

0.999

(4)

(5)

-

0.38 (0.15)

0.33 (0.14)

-

0.73 (0.15)

0.28 (0.21)

1.07 (0.01)

-

0.48 (0.15)

-

-0.19 (0.08)

-0.13 (0.19)

-

0.14 (0.08)

0.25 (0.19)

-0.19 (0.06) 0.998

0.999

-0.33 (0.19) 0.999

With the current values of income and price, and their values lagged one year, we see that lagged income has a higher coefficient than current income. This is plausible, since we expect inertia in the response. 27

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Alternative Dynamic Specifications, Housing Services Variable LGDPI

(1) 1.13 (0.01)

LGDPI(-1)

-

LGDPI(-2)

-

LGPRHOUS

-0.23 (0.07)

(2)

(3)

1.10 (0.01) -

LGPRHOUS(-1)

-

-0.20 (0.06)

LGPRHOUS(-2)

-

-

R2

0.998

0.999

(4)

(5)

-

0.38 (0.15)

0.33 (0.14)

-

0.73 (0.15)

0.28 (0.21)

1.07 (0.01)

-

0.48 (0.15)

-

-0.19 (0.08)

-0.13 (0.19)

-

0.14 (0.08)

0.25 (0.19)

-0.19 (0.06) 0.998

0.999

-0.33 (0.19) 0.999

However, the price side of the model does not behave in the same way. Indeed, one obtains a nonsensical positive coefficient for lagged price. 28

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Alternative Dynamic Specifications, Housing Services Variable

(1) (2) (3) Correlation Matrix =================================== LGDPI 1.13 LGDPI LGDPI(-1) (0.01) =================================== LGDPI(-1) 1.10 LGDPI 1.000000 0.998808 LGDPI(-1) 0.998808 (0.01) 1.000000 =================================== LGDPI(-2)

LGPRHOUS

-

-0.23 (0.07)

-

-

LGPRHOUS(-1)

-

-0.20 (0.06)

LGPRHOUS(-2)

-

-

R2

0.998

0.999

1.07 (0.01)

(4)

(5)

0.38 (0.15)

0.33 (0.14)

0.73 (0.15)

0.28 (0.21)

-

0.48 (0.15)

-

-0.19 (0.08)

-0.13 (0.19)

-

0.14 (0.08)

0.25 (0.19)

-0.19 (0.06) 0.998

0.999

-0.33 (0.19) 0.999

The problem, of course, is multcollinearity caused by the high correlation between current and lagged values. The correlation is particularly high for current and lagged income. 29

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Alternative Dynamic Specifications, Housing Services Variable

(1) (2) (3) Correlation Matrix =================================== LGDPI 1.13 LGPRHOUS LGPRHOUS(-1) (0.01) =================================== LGDPI(-1) 1.10 LGPRHOUS 1.000000 0.944448 LGPRHOUS(-1) 0.944448 (0.01) 1.000000 =================================== LGDPI(-2)

LGPRHOUS

-

-0.23 (0.07)

-

-

LGPRHOUS(-1)

-

-0.20 (0.06)

LGPRHOUS(-2)

-

-

R2

0.998

0.999

1.07 (0.01)

(4)

(5)

0.38 (0.15)

0.33 (0.14)

0.73 (0.15)

0.28 (0.21)

-

0.48 (0.15)

-

-0.19 (0.08)

-0.13 (0.19)

-

0.14 (0.08)

0.25 (0.19)

-0.19 (0.06) 0.998

0.999

-0.33 (0.19) 0.999

The correlation is also high for current and lagged price.

30

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Alternative Dynamic Specifications, Housing Services Variable LGDPI

(1) 1.13 (0.01)

LGDPI(-1)

-

LGDPI(-2)

-

LGPRHOUS

-0.23 (0.07)

(2)

(3)

1.10 (0.01) -

LGPRHOUS(-1)

-

-0.20 (0.06)

LGPRHOUS(-2)

-

-

R2

0.998

0.999

(4)

(5)

-

0.38 (0.15)

0.33 (0.14)

-

0.73 (0.15)

0.28 (0.21)

1.07 (0.01)

-

0.48 (0.15)

-

-0.19 (0.08)

-0.13 (0.19)

-

0.14 (0.08)

0.25 (0.19)

-0.19 (0.06) 0.998

0.999

-0.33 (0.19) 0.999

Notice how the standard errors have increased. The fact that the income coefficients seem plausible is probably just an accident. 31

TIME SERIES MODELS: STATIC MODELS AND MODELS WITH LAGS Alternative Dynamic Specifications, Housing Services Variable

(1) Matrix (2) (3) Correlation =============================================== LGDPI 1.13 LGDPI (0.01) LGDPI(-1) LGDPI(-2) =============================================== LGDPI(-1) 1.000000 - 0.998808 1.10 0.997614 LGDPI (0.01) 0.998774 LGDPI(-1) 0.998808 1.000000 LGDPI(-2) 1.000000 LGDPI(-2) 0.997614 - 0.998774 1.07 ===============================================

(4)

(5)

0.38 (0.15)

0.33 (0.14)

0.73 (0.15)

0.28 (0.21)

-

(0.01)

0.48 (0.15)

LGPRHOUS Correlation -0.23 Matrix (0.07) ===============================================

-0.19 (0.08)

-0.13 (0.19)

LGPRHOUS - LGPRHOUS(-1)LGPRHOUS(-2) LGPRHOUS(-1) -0.20 =============================================== (0.06)

0.14 (0.08)

0.25 (0.19)

LGPRHOUS 1.000000 0.944448 0.832581 LGPRHOUS(-2) -0.19 LGPRHOUS(-1) 0.944448 - 1.000000 0.944879 (0.06) LGPRHOUS(-2) 0.832581 0.944879 1.000000 =============================================== 2

R

0.998

0.999

0.998

0.999

-0.33 (0.19) 0.999

If we add income and price lagged two years, the results become even more erratic. For a category of expenditure such as housing, where one might expect long lags, this is clearly not a constructive approach to determining the lag structure. 32

ADAPTIVE EXPECTATIONS

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

Expectations are often important in economic models of dynamic processes, particularly in macroeconomic models, and finding ways to model them is often an important and difficult task for the applied economist using time series data. 1

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

The adaptive expectations model was one of the earliest approaches developed for this purpose. Suppose that you hypothesize that a variable y at time t is related, not to the actual value of an explanatory variable x, but to the value it is expected to have at time t+1. 2

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 − x te = λ ( x t − x te )

The expected value of x is unobservable. One solution to this problem is to hypothesize that, at time t, the value that had been expected for time t is compared with the actual value and the expected value for the next period is adjusted by a proportion λ of the discrepancy. 3

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 − x te = λ ( x t − x te ) x te+1 = λx t + (1 − λ ) x te

The adaptive expectations model thus implies that the expectation for time period t+1 is a weighted average of the actual outcome for time t and the value that had been expected for time t. 4

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 − x te = λ ( x t − x te ) x te+1 = λx t + (1 − λ ) x te

λ should lie in the range 0 to 1.

The higher its value, the more rapidly do expectations change in response to outcomes. At the extreme value of 1, adjustment is instantaneous and we are back with the static model. At the other extreme, 0, there is no response at all. 5

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 − x te = λ ( x t − x te )

yt = α + β (λx t + (1 − λ ) x ) + ut e t

x te+1 = λx t + (1 − λ ) x te

= α + βλx t + β (1 − λ ) x te + ut

To obtain a model in which y is related to observable variables, the first step is to substitute for the expected value of x using the adaptive expectations equation. 6

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 − x te = λ ( x t − x te )

yt = α + β (λx t + (1 − λ ) x ) + ut e t

= α + βλx t + β (1 − λ ) x te + ut

x te+1 = λx t + (1 − λ ) x te x te = λx t −1 + (1 − λ ) x te−1

Of course, we still have an unobservable variable on the right side of the equation, but we can deal with this by returning to the adaptive expectations model. If the process is true for time t+1, it is also true for time t. 7

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 − x te = λ ( x t − x te )

yt = α + β (λx t + (1 − λ ) x ) + ut e t

= α + βλx t + β (1 − λ ) x te + ut

x te+1 = λx t + (1 − λ ) x te x te = λx t −1 + (1 − λ ) x te−1

yt = α + βλx t + βλ (1 − λ ) x t −1 + β (1 − λ ) 2 x te−1 + ut

We now have y as a function of xt, xt-1, and the value of x expected at time t-1. The latter is unobservable, but we substitute for it from the adaptive expectations process, lagged one further period. 8

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 − x te = λ ( x t − x te )

yt = α + β (λx t + (1 − λ ) x ) + ut e t

= α + βλx t + β (1 − λ ) x te + ut

x te+1 = λx t + (1 − λ ) x te x te = λx t −1 + (1 − λ ) x te−1

yt = α + βλx t + βλ (1 − λ ) x t −1 + β (1 − λ ) 2 x te−1 + ut yt = α + βλxt + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 xt −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s +1 + ut

After lagging the adaptive expectations process and substituting s times, we arrive at the equation shown. 9

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 − x te = λ ( x t − x te )

yt = α + β (λx t + (1 − λ ) x ) + ut e t

= α + βλx t + β (1 − λ ) x te + ut

x te+1 = λx t + (1 − λ ) x te x te = λx t −1 + (1 − λ ) x te−1

yt = α + βλx t + βλ (1 − λ ) x t −1 + β (1 − λ ) 2 x te−1 + ut yt = α + βλxt + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 xt −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s +1 + ut

Now, if λ lies between 0 and 1, so will (1- λ), and hence the term (1- λ)s is a diminishing function of s. For s large enough, the coefficient of the last term will be so small that the term can be dropped and we have expressed y exclusively in terms of observable variables. 10

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut yˆ t = 101 + 60 x t + 45 x t −1 + 20 x t −2

You would not use OLS to fit this model. If you did, the first four terms of the regression equation might be as shown. 11

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut yˆ t = 101 + 60 x t + 45 x t −1 + 20 x t −2 bl = 60, bl (1 − l ) = 45, bl (1 − l ) 2 = 20

Relating the numerical estimates to the true model, we obtain three equations involving b and l. Solving the first two equations, l = 0.25 and b = 240. However these values are incompatible with the third equation. 12

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut yˆ t = 101 + 60 x t + 45 x t −1 + 20 x t −2 bl = 60, bl (1 − l ) = 45, bl (1 − l ) 2 = 20 l = 0.25, b = 240

OLS estimators remain consistent, so if the the sample were very large there would be no contradictions, but for a finite sample there would be an incompatibility problem. 13

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut yˆ t = 101 + 60 x t + 45 x t −1 + 20 x t −2 bl = 60, bl (1 − l ) = 45, bl (1 − l ) 2 = 20 l = 0.25, b = 240

The problem would be made worse by the fact that the model would suffer from multicollinearity because the current and lagged time series for x are likely to be highly correlated. 14

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut

The solution to this problem is to use a nonlinear estimation technique that takes account of the relationships among the coefficients. Nonlinear estimation is now a standard feature on most serious regression applications. 15

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut

If your regression application does not allow nonlinear estimation, you could still fit this model using a grid search. We will look briefly at this technique, despite the fact that it is obsolete, because it makes clear that the problem of multicollinearity has been solved. 16

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut yt = α + βz t + ut z t = λ x t + λ (1 − λ ) x t −1 + λ (1 − λ ) 2 x t −2 + ... + λ (1 − λ ) s−1 x t − s +1 + (1 − λ ) s x te− s +1

We split the model into two equations, as shown. The values of the z series depend of course on the value of λ. You construct 10 versions of the z series using the following values for λ : 0.1, 0.2, ..., 1.0, and fit the middle equation for each of them. 17

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut yt = α + βz t + ut z t = λ x t + λ (1 − λ ) x t −1 + λ (1 − λ ) 2 x t −2 + ... + λ (1 − λ ) s−1 x t − s +1 + (1 − λ ) s x te− s +1

The version with the lowest residual sum of squares is, by definition, the least squares solution. Note that these regressions are simple regressions of y on z and so multicollinearity has been eliminated. 18

ADAPTIVE EXPECTATIONS

λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b1 s.e.(b1) 1.67 1.22 1.13 1.12 1.13 1.15 1.16 1.17 1.17 1.18

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

b2 s.e.(b2) -0.35 -0.28 -0.28 -0.30 -0.32 -0.34 -0.36 -0.38 -0.39 -0.40

0.07 0.04 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.05

RSS 0.001636 0.001245 0.000918 0.000710 0.000666 0.000803 0.001109 0.001561 0.002137 0.002823

Here is the result of a grid search for a logarithmic regression of expenditure on housing services on income and price using data from the Demand Functions data set. Eight lagged values were used. 19

ADAPTIVE EXPECTATIONS

λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b1 s.e.(b1) 1.67 1.22 1.13 1.12 1.13 1.15 1.16 1.17 1.17 1.18

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

b2 s.e.(b2) -0.35 -0.28 -0.28 -0.30 -0.32 -0.34 -0.36 -0.38 -0.39 -0.40

0.07 0.04 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.05

RSS 0.001636 0.001245 0.000918 0.000710 0.000666 0.000803 0.001109 0.001561 0.002137 0.002823

RSS is minimized for λ = 0.5, implying that expectations are adjusted by half of the discrepancy between actual and expected values in the current year. 20

ADAPTIVE EXPECTATIONS

λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b1 s.e.(b1) 1.67 1.22 1.13 1.12 1.13 1.15 1.16 1.17 1.17 1.18

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

b2 s.e.(b2) -0.35 -0.28 -0.28 -0.30 -0.32 -0.34 -0.36 -0.38 -0.39 -0.40

0.07 0.04 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.05

RSS 0.001636 0.001245 0.000918 0.000710 0.000666 0.000803 0.001109 0.001561 0.002137 0.002823

We could obtain a more accurate estimate of λ and the other parameters by performing a further grid search for values of λ between 0.4 and 0.6 with steps of 0.01. 21

ADAPTIVE EXPECTATIONS

λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b1 s.e.(b1) 1.67 1.22 1.13 1.12 1.13 1.15 1.16 1.17 1.17 1.18

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

b2 s.e.(b2) -0.35 -0.28 -0.28 -0.30 -0.32 -0.34 -0.36 -0.38 -0.39 -0.40

0.07 0.04 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.05

RSS 0.001636 0.001245 0.000918 0.000710 0.000666 0.000803 0.001109 0.001561 0.002137 0.002823

The estimates of the income and price elasticities are not very different from those in the static model (λ = 1.0). 22

ADAPTIVE EXPECTATIONS

λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b1 s.e.(b1) 1.67 1.22 1.13 1.12 1.13 1.15 1.16 1.17 1.17 1.18

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

b2 s.e.(b2) -0.35 -0.28 -0.28 -0.30 -0.32 -0.34 -0.36 -0.38 -0.39 -0.40

0.07 0.04 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.05

RSS 0.001636 0.001245 0.000918 0.000710 0.000666 0.000803 0.001109 0.001561 0.002137 0.002823

(If you have noticed that the static model estimates are different from those in the previous sequence, that is because the sample period has been shortened to 1967-1994, to allow eight lags.) 23

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut yt = α + βx te+1 + ut x te+1 − x te = λ ( x t − x te )

In this model the short-run impact of x on y is given by βλ. This is the coefficient of xt. In time period t, the previous values of x are fixed. We will demonstrate that the long-run impact, comparing equilibria, is given by β. 24

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut yt = α + βx te+1 + ut x te+1 − x te = λ ( x t − x te )

It is obvious intuitively that this should be the case. In equilibrium, there is no discrepancy between the actual and expected value of x, and hence in the second equation the expected value can be replaced by the actual value. 25

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut x t = xt −1 = x t −2 = ... = x

If the process were in equilibrium, all the values of x would be equal to some equilibrium value, x. 26

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut x t = xt −1 = x t −2 = ... = x y = α + βλx + βλ (1 − λ ) x + βλ (1 − λ ) 2 x + ... = α + β x (λ + λ (1 − λ ) + λ (1 − λ ) 2 + ...) =α + β x

Substituting these values into the first equation, we obtain an expression for the equilibrium value of y. 27

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut x t = xt −1 = x t −2 = ... = x y = α + βλx + βλ (1 − λ ) x + βλ (1 − λ ) 2 x + ... = α + β x (λ + λ (1 − λ ) + λ (1 − λ ) 2 + ...) =α + β x

We group the terms together.

28

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut x t = xt −1 = x t −2 = ... = x y = α + βλx + βλ (1 − λ ) x + βλ (1 − λ ) 2 x + ... = α + β x (λ + λ (1 − λ ) + λ (1 − λ ) 2 + ...) =α + β x

Thus we have demonstrated that β gives the impact of the equilibrium value of x on the equilibrium value of y. 29

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut x t = xt −1 = x t −2 = ... = x y = α + βλx + βλ (1 − λ ) x + βλ (1 − λ ) 2 x + ... = α + β x (λ + λ (1 − λ ) + λ (1 − λ ) 2 + ...) =α + β x

S = λ + λ (1 − λ ) + λ (1 − λ ) 2 + ... (1 − λ ) S = λ (1 − λ ) + λ (1 − λ ) 2 + ... S − (1 − λ ) S = λ λS = λ

Possibly it was not obvious that the term involving the λs is equal to 1. We will demonstrate this. Call the term S. 30

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut x t = xt −1 = x t −2 = ... = x y = α + βλx + βλ (1 − λ ) x + βλ (1 − λ ) 2 x + ... = α + β x (λ + λ (1 − λ ) + λ (1 − λ ) 2 + ...) =α + β x

S = λ + λ (1 − λ ) + λ (1 − λ ) 2 + ... (1 − λ ) S = λ (1 − λ ) + λ (1 − λ ) 2 + ... S − (1 − λ ) S = λ λS = λ

Multiply both sides by (1 - λ).

31

ADAPTIVE EXPECTATIONS

yt = α + βλx t + βλ (1 − λ ) x t −1 + βλ (1 − λ ) 2 x t −2 + ... + βλ (1 − λ ) s −1 x t − s +1 + β (1 − λ ) s x te− s+1 + ut x t = xt −1 = x t −2 = ... = x y = α + βλx + βλ (1 − λ ) x + βλ (1 − λ ) 2 x + ... = α + β x (λ + λ (1 − λ ) + λ (1 − λ ) 2 + ...) =α + β x

S = λ + λ (1 − λ ) + λ (1 − λ ) 2 + ... (1 − λ ) S = λ (1 − λ ) + λ (1 − λ ) 2 + ... S − (1 − λ ) S = λ λS = λ

Subtract the second equation from the first. On simplifying, S is equal to 1.

32

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 = λx t + (1 − λ ) x te

yt = α + β (λx t + (1 − λ ) x te ) + ut = α + βλx t + β (1 − λ ) x te + ut

We have seen that the model implies different short-run and long-run effects of x on y. We will illustrate these graphically. 33

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 = λx t + (1 − λ ) x te

yt = α + β (λx t + (1 − λ ) x te ) + ut = α + βλx t + β (1 − λ ) x te + ut

First, we need to manipulate the model a little. We start off as before, substituting for the expected value of x in period t+1 in the first equation. 34

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 = λx t + (1 − λ ) x te

yt = α + β (λx t + (1 − λ ) x te ) + ut = α + βλx t + β (1 − λ ) x te + ut yt −1 = α + βx te + ut −1

Now we write the first equation a second time, lagging it one period.

35

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 = λx t + (1 − λ ) x te

yt = α + β (λx t + (1 − λ ) x te ) + ut = α + βλx t + β (1 − λ ) x te + ut yt −1 = α + βx te + ut −1

βxte = yt −1 − α − ut −1

We have rearranged the equation so that the expected value of x at time t is on the left side.

36

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 = λx t + (1 − λ ) x te

yt = α + β (λx t + (1 − λ ) x te ) + ut = α + βλx t + β (1 − λ ) x te + ut yt −1 = α + βx te + ut −1

βxte = yt −1 − α − ut −1 yt = α + βλx t + (1 − λ )( yt −1 − α − ut −1 ) + ut = αλ + (1 − λ ) yt −1 + βλxt + ut − (1 − λ )ut −1 We use this equation to eliminate the expected value of x from the equation for y.

37

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 = λx t + (1 − λ ) x te

yt = α + β (λx t + (1 − λ ) x te ) + ut = α + βλx t + β (1 − λ ) x te + ut yt −1 = α + βx te + ut −1

βxte = yt −1 − α − ut −1 yt = α + βλx t + (1 − λ )( yt −1 − α − ut −1 ) + ut = αλ + (1 − λ ) yt −1 + βλxt + ut − (1 − λ )ut −1 Hence we obtain y as a function of lagged y, current x, and a compound disturbance term.

38

ADAPTIVE EXPECTATIONS

yt = α + βx te+1 + ut

x te+1 = λx t + (1 − λ ) x te

yt = α + β (λx t + (1 − λ ) x te ) + ut = α + βλx t + β (1 − λ ) x te + ut yt −1 = α + βx te + ut −1

βxte = yt −1 − α − ut −1 yt = α + βλx t + (1 − λ )( yt −1 − α − ut −1 ) + ut = αλ + (1 − λ ) yt −1 + βλxt + ut − (1 − λ )ut −1 First, we will investigate the short-run effect of x on y. The impact of current x on y is given by βλ, as we saw before. At time t, yt-1 has already been determined, so the intercept for the short-run function is (αλ + [1 - λ]yt-1).

39

ADAPTIVE EXPECTATIONS

y t = αλ + (1- λ )y t -1 + βλx t + ut - (1- λ )ut -1 y

αλ + (1-λ)yt-1 xt

x

The short-run relationship is depicted graphically. The slope of the line is βλ.and the intercept is (αλ + [1 - λ]yt-1) 40

ADAPTIVE EXPECTATIONS

y t +1 = αλ + (1- λ )y t + βλx t +1 + ut +1 - (1- λ )ut y

αλ + (1-λ)yt αλ + (1-λ)yt-1 xt

xt+1

x

For simplicity, we will suppose that x increases with time. If this is the case, y will also increase with time, for two reasons. It will increase because x is increasing, but it will also increase because the intercept, being a function of y, will shift upwards. 41

ADAPTIVE EXPECTATIONS

y

αλ + (1-λ)yt+1 αλ + (1-λ)yt αλ + (1-λ)yt-1 xt

xt+1 xt+2

x

The effect of the second component is to make the actual relationship between y and x different from that implied by the short-run relationships depicted by the lines. 42

ADAPTIVE EXPECTATIONS

Long run

y

αλ + (1-λ)yt+3 αλ + (1-λ)yt+2 αλ + (1-λ)yt+1 Short run

αλ + (1-λ)yt αλ + (1-λ)yt-1 xt

xt+1 xt+2

xt+3

xt+4

x

In this example, it is steeper.

43

ADAPTIVE EXPECTATIONS

yt = αλ + (1 − λ ) yt −1 + βλxt + ut − (1 − λ )ut −1

λyt ≈ αλ + βλxt y t = α + βx t

To determine the long-run relationship, we use the fact that yt-1 will be similar to yt and hence, neglecting the effect of the disturbance term, we can rewrite the equation as shown. 44

ADAPTIVE EXPECTATIONS

yt = αλ + (1 − λ ) yt −1 + βλxt + ut − (1 − λ )ut −1

λyt ≈ αλ + βλxt y t = α + βx t

Thus again we see that the long-run effect is given by β.

45

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P C t = C tP + C tT Yt = Yt P + YtT

In the years after the Second World War, econometricians working with macroeconomic data were puzzled by the fact that the long-run average propensity to consume seemed to be roughly constant despite the marginal propensity to consume being much lower. 1

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P C t = C tP + C tT Yt = Yt P + YtT

A model in which current consumption was a function of current income could not explain this phenomenon and was therefore clearly too simplistic. 2

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P C t = C tP + C tT Yt = Yt P + YtT

Several more sophisticated models were developed which could explain this apparent contradiction, one of them being Friedman's Permanent Income Hypothesis. 3

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P C t = C tP + C tT Yt = Yt P + YtT

According to the Permanent Income Hypothesis, permanent consumption, CP, is proportional to permanent income, YP. Permanent income is a subjective notion of likely medium-run future income. Permanent consumption is a similar notion for consumption. 4

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P C t = C tP + C tT Yt = Yt P + YtT

Actual consumption, C, and actual income, Y, consist of these permanent components plus unanticipated transitory components, CT and YT, respectively. 5

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P C t = C tP + C tT Yt = Yt P + YtT

It is assumed, at least as a first approximation, that the transitory components of consumption and income have expected value 0 and are distributed independently of their permanent counterparts and of each other. 6

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT Yt = Yt P + YtT

To solve the problem that permanent income is unobservable, Friedman hypothesized that it is subject to an adaptive expectations process, permanent income at time t being updated by a proportion of the difference between actual income and permanent income at time t-1. 7

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Hence permanent income at time t is a weighted average of actual income at time t and permanent income at time t-1. 8

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT C t − C tT = β (λYt + (1 − λ )Yt −P1 )

This relationship can be used to substitute for permanent income in the relationship between permanent consumption and permanent income. We have also substituted for permanent consumption using the identity for actual consumption. 9

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT C t − C tT = β (λYt + (1 − λ )Yt −P1 ) C t = βλYt + β (1 − λ )Yt −P1 + C tT

Thus we obtain current consumption as a function of current income and permanent income lagged one period. 10

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Yt −P1 = λYt −1 + (1 − λ )Yt −P2

C t − C tT = β (λYt + (1 − λ )Yt −P1 ) C t = βλYt + β (1 − λ )Yt −P1 + C tT C t = βλYt + βλ (1 − λ )Yt −1 + β (1 − λ ) 2 Yt −P2 + C tT

The latter is unobservable, but it can be eliminated by lagging the adaptive expectations process and substituting. Of course, we still have an unobservable variable, permanent income lagged two periods, on the right side of the equation. 11

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Yt −P1 = λYt −1 + (1 − λ )Yt −P2

C t − C tT = β (λYt + (1 − λ )Yt −P1 ) C t = βλYt + β (1 − λ )Yt −P1 + C tT C t = βλYt + βλ (1 − λ )Yt −1 + β (1 − λ ) 2 Yt −P2 + C tT C t = β λYt + βλ (1 − λ )Yt −1 + βλ (1 − λ ) 2 Yt −2 + ... + βλ (1 − λ ) s−1 Yt − s+1 + β (1 − λ ) s Yt −Ps+1 + C tT After lagging and substituting s times, we obtain the equation shown. It is reasonable to assume that λ lies between 0 and 1, and so (1 - λ) will also lie between 0 and 1. (1 - λ)s will therefore be a declining function of s, and for s large enough the final term can be dropped. 12

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Yt −P1 = λYt −1 + (1 − λ )Yt −P2

C t − C tT = β (λYt + (1 − λ )Yt −P1 ) C t = βλYt + β (1 − λ )Yt −P1 + C tT C t = βλYt + βλ (1 − λ )Yt −1 + β (1 − λ ) 2 Yt −P2 + C tT C t = β λYt + βλ (1 − λ )Yt −1 + βλ (1 − λ ) 2 Yt −2 + ... + βλ (1 − λ ) s−1 Yt − s+1 + β (1 − λ ) s Yt −Ps+1 + C tT Friedman fitted this model using a grid search on the lines of that described in the previous sequence. 13

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Yt −P1 = λYt −1 + (1 − λ )Yt −P2

C t − C tT = β (λYt + (1 − λ )Yt −P1 ) C t = βλYt + β (1 − λ )Yt −P1 + C tT

To see how this model reconciles a low short-run marginal propensity to consume with a higher long-run average propensity, it is convenient to manipulate it a little. We start with actual consumption written as a function of actual income and lagged permanent income. 14

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Yt −P1 = λYt −1 + (1 − λ )Yt −P2

C t − C tT = β (λYt + (1 − λ )Yt −P1 ) C t = βλYt + β (1 − λ )Yt −P1 + C tT

βYt −P1 = C tP−1 = C t −1 − C tT−1

Now we make use of the basic relationship between permanent consumption and permanent income, lagging it one period. 15

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Yt −P1 = λYt −1 + (1 − λ )Yt −P2

C t − C tT = β (λYt + (1 − λ )Yt −P1 ) C t = βλYt + β (1 − λ )Yt −P1 + C tT

βYt −P1 = C tP−1 = C t −1 − C tT−1 C t = βλYt + (1 − λ )(C t −1 − C tT−1 ) + C tT = βλYt + (1 − λ )C t −1 + C tT − (1 − λ )C tT−1 This allows us to eliminate lagged permanent income from the consumption function.

16

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Yt −P1 = λYt −1 + (1 − λ )Yt −P2

C t − C tT = β (λYt + (1 − λ )Yt −P1 ) C t = βλYt + β (1 − λ )Yt −P1 + C tT

βYt −P1 = C tP−1 = C t −1 − C tT−1 C t = βλYt + (1 − λ )(C t −1 − C tT−1 ) + C tT = βλYt + (1 − λ )C t −1 + C tT − (1 − λ )C tT−1 Thus we obtain a model with no unobservable variables. The short-run marginal propensity to consume is given by the coefficient of Yt, βλ. 17

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Yt −P1 = λYt −1 + (1 − λ )Yt −P2

C t = βλYt + (1 − λ )C t −1 + C tT − (1 − λ )C tT−1

C = βλY + (1 − λ )C

To obtain a measure of the long-run propensity, we perform an exercise in comparative statics, seeing how equilibrium consumption is related to equilibrium income in this model. 18

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C tP = βYt P

Yt P − Yt −P1 = λ (Yt − Yt −P1 )

C t = C tP + C tT

Yt P = λYt + (1 − λ )Yt −P1

Yt = Yt P + YtT

Yt −P1 = λYt −1 + (1 − λ )Yt −P2

C t = βλYt + (1 − λ )C t −1 + C tT − (1 − λ )C tT−1 C = βλY + (1 − λ )C

λC = βλY C = βY

The long-run propensity is β. This makes sense intuitively. The fundamental relationship is that between permanent consumption and permanent income, with coefficient β. The transitory components are responsible for short-run deviations from this relationship. 19

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C

(1-λ)Ct+3 (1-λ)Ct+2 (1-λ)Ct+1 (1-λ)Ct (1-λ)Ct-1 Yt

Yt+1 Yt+2

Yt+3

Yt+4

Y

Here is the dynamics diagram from the previous sequence, reinterpreted as Friedman's model. The slope of the short-run function is βλ. 20

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C

(1-λ)Ct+3 (1-λ)Ct+2 (1-λ)Ct+1 (1-λ)Ct (1-λ)Ct-1 Yt

Yt+1 Yt+2

Yt+3

Yt+4

Y

The intercept of the short-run function at time t is (1 - λ)Ct-1. If income is increasing, it shifts upwards, and the actual long-run relationship between C and Y is steeper, the slope approximating β. 21

ADAPTIVE EXPECTATIONS: FRIEDMAN'S PERMANENT INCOME HYPOTHESIS

C

(1-λ)Ct+3 (1-λ)Ct+2 (1-λ)Ct+1 (1-λ)Ct (1-λ)Ct-1 Yt

Yt+1 Yt+2

Yt+3

Yt+4

Y

Thus the model reconciles a low short-run marginal propensity to consume with a high long-run average propensity. 22

PARTIAL ADJUSTMENT

PARTIAL ADJUSTMENT

yt* = α + βx t + ut

The partial adjustment model is another simple dynamic process common in models using time series data. In this model it is assumed that the desired, or target, value of the dependent variable, yt*, is related to one or more explanatory variables. 1

PARTIAL ADJUSTMENT

yt* = α + βx t + ut

yt − yt −1 = λ ( yt* − yt −1 )

However, the actual value of the dependent variable is different. It lags behind the desired value, the increase in it in any time period being a proportion of the discrepancy between the desired value and the actual value in the previous time period. 2

PARTIAL ADJUSTMENT

yt* = α + βx t + ut

yt − yt −1 = λ ( yt* − yt −1 ) yt = λyt* + (1 − λ ) yt −1

The actual value in the current time period is therefore a weighted average of the desired value and the previous actual value. λ logically should lie in the interval 0 (no change at all) to 1 (full adjustment in the current time period). 3

PARTIAL ADJUSTMENT

yt* = α + βx t + ut

yt − yt −1 = λ ( yt* − yt −1 ) yt = λyt* + (1 − λ ) yt −1

yt = λ (α + βx t + ut ) + (1 − λ ) yt −1 = αλ + βλx t + (1 − λ ) yt −1 + λut

Substituting for the desired value from the original relationship, one obtains a relationship with only observable variables. Thus the parameters α, β, and λ can be estimated by regressing yt on xt and yt-1. 4

PARTIAL ADJUSTMENT

yt* = α + βx t + ut

yt − yt −1 = λ ( yt* − yt −1 ) yt = λyt* + (1 − λ ) yt −1

yt = λ (α + βx t + ut ) + (1 − λ ) yt −1 = αλ + βλx t + (1 − λ ) yt −1 + λut

The model relates y to the current value of x and the lagged value of itself, and thus has the same structure as the adaptive expectations model. 5

PARTIAL ADJUSTMENT

yt* = α + βx t + ut

yt − yt −1 = λ ( yt* − yt −1 ) yt = λyt* + (1 − λ ) yt −1

yt = λ (α + βx t + ut ) + (1 − λ ) yt −1 = αλ + βλx t + (1 − λ ) yt −1 + λut

It follows that its dynamics are exactly the same. The short-run impact of x on y is given by the coefficient βλ. 6

PARTIAL ADJUSTMENT

yt* = α + βx t + ut

yt − yt −1 = λ ( yt* − yt −1 ) yt = λyt* + (1 − λ ) yt −1

yt = λ (α + βx t + ut ) + (1 − λ ) yt −1 = αλ + βλx t + (1 − λ ) yt −1 + λut y = αλ + βλ x + (1 − λ ) y

The long-run effect can be evaluated by finding the relationship between the equilibrium values of y and x. 7

PARTIAL ADJUSTMENT

yt* = α + βx t + ut

yt − yt −1 = λ ( yt* − yt −1 ) yt = λyt* + (1 − λ ) yt −1

yt = λ (α + βx t + ut ) + (1 − λ ) yt −1 = αλ + βλx t + (1 − λ ) yt −1 + λut y = αλ + βλ x + (1 − λ ) y

λy = αλ + βλ x y = α + βx The long-run effect turns out to be β. This makes sense, since this is the coefficient in the equation determining the desired value of y. 8

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

Brown's Habit Persistence Model of the aggregate consumption function was an early example of the use of a partial adjustment model. Desired consumption is related to wage income, nonwage income and a dummy variable. 9

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

The reason for separating income into wage income and nonwage income is that the marginal propensity to consume is likely to be higher for wage income than for nonwage income. 10

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

Brown fitted the model with a time series which included observations before and after the Second World War. The dummy variable, A, was defined to be 0 for the prewar observations and 1 for the postwar ones. 11

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

C t − C t −1 = λ (C t* − C t −1 ) C t = λC t* + (1 − λ )C t −1

As the name of his model suggests, Brown hypothesized that there was a lag in the response of consumption to changes in income and he used a partial adjustment model. 12

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

C t − C t −1 = λ (C t* − C t −1 ) C t = λC t* + (1 − λ )C t −1

C t = λ (α + β 1Wt + β 2 NWt + A + ut ) + (1 − λ )C t −1 = αλ + β 1λWt + β 2 λNWt + (1 − λ )C t −1 + λA + λut

Substituting for desired consumption, one obtains current consumption in terms of current income and previous consumption. 13

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

C t − C t −1 = λ (C t* − C t −1 ) C t = λC t* + (1 − λ )C t −1

C t = λ (α + β 1Wt + β 2 NWt + A + ut ) + (1 − λ )C t −1 = αλ + β 1λWt + β 2 λNWt + (1 − λ )C t −1 + λA + λut Cˆ t = 0.90 + 0.61Wt + 0.28 NWt + 0.22C t −1 + 0.69 A (4.8) (7.4) (4.2) (2.8) (4.8)

Brown fitted the model with aggregate Canadian data for the years 1926-1949, omitting the years 1942-1945, using a simultaneous equations estimation technique. The variables were measured in billions of Canadian dollars at constant prices. t statistics are in parentheses. 14

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

C t − C t −1 = λ (C t* − C t −1 ) C t = λC t* + (1 − λ )C t −1

C t = λ (α + β 1Wt + β 2 NWt + A + ut ) + (1 − λ )C t −1 = αλ + β 1λWt + β 2 λNWt + (1 − λ )C t −1 + λA + λut Cˆ t = 0.90 + 0.61Wt + 0.28 NWt + 0.22C t −1 + 0.69 A (4.8) (7.4) (4.2) (2.8) (4.8)

The short-run marginal propensities to consume out of wage and nonwage income are 0.61 and 0.28, respectively. Note that the former is indeed larger than the latter. How would you test whether the difference is significant? 15

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

C t − C t −1 = λ (C t* − C t −1 ) C t = λC t* + (1 − λ )C t −1

C t = λ (α + β 1Wt + β 2 NWt + A + ut ) + (1 − λ )C t −1 = αλ + β 1λWt + β 2 λNWt + (1 − λ )C t −1 + λA + λut Cˆ t = 0.90 + 0.61Wt + 0.28 NWt + 0.22C t −1 + 0.69 A (4.8) (7.4) (4.2) (2.8) (4.8)

The coefficient of lagged consumption literally implies that, if consumption in the previous year had been 1 billion dollars greater, consumption this year would have been 0.22 billion dollars greater. 16

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

C t − C t −1 = λ (C t* − C t −1 ) C t = λC t* + (1 − λ )C t −1

C t = λ (α + β 1Wt + β 2 NWt + A + ut ) + (1 − λ )C t −1 = αλ + β 1λWt + β 2 λNWt + (1 − λ )C t −1 + λA + λut Cˆ t = 0.90 + 0.61Wt + 0.28 NWt + 0.22C t −1 + 0.69 A (4.8) (7.4) (4.2) (2.8) (4.8)

That is a bit clumsy. It is better to interpret it with reference to λ in the adjustment process. It implies that the speed of adjustment is 0.78, meaning that 0.78 of the difference between desired and actual consumption is eliminated in one year. 17

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

C t − C t −1 = λ (C t* − C t −1 ) C t = λC t* + (1 − λ )C t −1

C t = λ (α + β 1Wt + β 2 NWt + A + ut ) + (1 − λ )C t −1 = αλ + β 1λWt + β 2 λNWt + (1 − λ )C t −1 + λA + λut Cˆ t = 0.90 + 0.61Wt + 0.28 NWt + 0.22C t −1 + 0.69 A (4.8) (7.4) (4.2) (2.8) (4.8)

0.61 b1 = = 0.78 1 − 0.22

0.28 b2 = = 0.36 1 − 0.22

With the speed of adjustment, we can derive the long-run propensities to consume. We do this by dividing the short-run propensities by λ. We find that the long-run propensity to consume out of wages is 0.78. 18

PARTIAL ADJUSTMENT

C t* = α + β 1Wt + β 2 NWt + δA + ut

C t − C t −1 = λ (C t* − C t −1 ) C t = λC t* + (1 − λ )C t −1

C t = λ (α + β 1Wt + β 2 NWt + A + ut ) + (1 − λ )C t −1 = αλ + β 1λWt + β 2 λNWt + (1 − λ )C t −1 + λA + λut Cˆ t = 0.90 + 0.61Wt + 0.28 NWt + 0.22C t −1 + 0.69 A (4.8) (7.4) (4.2) (2.8) (4.8)

0.61 b1 = = 0.78 1 − 0.22

0.28 b2 = = 0.36 1 − 0.22

Similarly, the long-run propensity to consume nonwage income is 0.36. Note that, in this example, there is not a great difference between the short-run and long-run propensities. That is because the speed of adjustment is rapid. 19

PARTIAL ADJUSTMENT

============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 1994 Included observations: 35 after adjusting endpoints ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -0.390249 0.152989 -2.550839 0.0159 LGDPI 0.313919 0.052510 5.978243 0.0000 LGPRHOUS -0.067547 0.024689 -2.735882 0.0102 LGHOUS(-1) 0.701432 0.045082 15.55895 0.0000 ============================================================== R-squared 0.999773 Mean dependent var 6.017555 Adjusted R-squared 0.999751 S.D. dependent var 0.362063 S.E. of regression 0.005718 Akaike info criter -10.22102 Sum squared resid 0.001014 Schwarz criterion -10.04327 Log likelihood 133.2051 F-statistic 45427.98 Durbin-Watson stat 1.718168 Prob(F-statistic) 0.000000 ============================================================== Here is the result of a parallel logarithmic regression of expenditure on housing on DPI and relative price, using the Demand Functions data set. 20

PARTIAL ADJUSTMENT

============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 1994 Included observations: 35 after adjusting endpoints ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -0.390249 0.152989 -2.550839 0.0159 LGDPI 0.313919 0.052510 5.978243 0.0000 LGPRHOUS -0.067547 0.024689 -2.735882 0.0102 LGHOUS(-1) 0.701432 0.045082 15.55895 0.0000 ============================================================== R-squared 0.999773 Mean dependent var 6.017555 Adjusted R-squared 0.999751 S.D. dependent var 0.362063 S.E. of regression 0.005718 Akaike info criter -10.22102 Sum squared resid 0.001014 Schwarz criterion -10.04327 Log likelihood 133.2051 F-statistic 45427.98 Durbin-Watson stat 1.718168 Prob(F-statistic) 0.000000 ============================================================== The short-run income elasticity is 0.32.

21

PARTIAL ADJUSTMENT

============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 1994 Included observations: 35 after adjusting endpoints ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -0.390249 0.152989 -2.550839 0.0159 LGDPI 0.313919 0.052510 5.978243 0.0000 LGPRHOUS -0.067547 0.024689 -2.735882 0.0102 LGHOUS(-1) 0.701432 0.045082 15.55895 0.0000 ============================================================== R-squared 0.999773 Mean dependent var 6.017555 Adjusted R-squared 0.999751 S.D. dependent var 0.362063 S.E. of regression 0.005718 Akaike info criter -10.22102 Sum squared resid 0.001014 Schwarz criterion -10.04327 Log likelihood 133.2051 F-statistic 45427.98 Durbin-Watson stat 1.718168 Prob(F-statistic) 0.000000 ============================================================== The short-run price elasticity is 0.07. Both of these elasticities are very low. This is because housing is a good example of a category of expenditure with slow adjustment. 22

PARTIAL ADJUSTMENT

============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 1994 Included observations: 35 after adjusting endpoints ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -0.390249 0.152989 -2.550839 0.0159 LGDPI 0.313919 0.052510 5.978243 0.0000 LGPRHOUS -0.067547 0.024689 -2.735882 0.0102 LGHOUS(-1) 0.701432 0.045082 15.55895 0.0000 ============================================================== R-squared 0.999773 Mean dependent var 6.017555 Adjusted R-squared 0.999751 S.D. dependent var 0.362063 S.E. of regression 0.005718 Akaike info criter -10.22102 Sum squared resid 0.001014 Schwarz criterion -10.04327 Log likelihood 133.2051 F-statistic 45427.98 Durbin-Watson stat 1.718168 Prob(F-statistic) 0.000000 ============================================================== The adjustment rate implicit in the coefficient of LGHOUS(-1) is only 0.30. People do not change their housing quickly in response to changes in income and price. If anything, the estimated rate seems a little high. 23

PARTIAL ADJUSTMENT

============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 1994 Included observations: 35 after adjusting endpoints ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -0.390249 0.152989 -2.550839 0.0159 LGDPI 0.313919 0.052510 5.978243 0.0000 LGPRHOUS -0.067547 0.024689 -2.735882 0.0102 LGHOUS(-1) 0.701432 0.045082 15.55895 0.0000 ==============================================================

0.3139 b1 = = 1.05 1 − 0.7014

0.0675 b2 = = 0.23 1 − 0.7014

The long-run income elasticity is 1.05, not far off the income elasticity in the static model in the first sequence for this chapter. 24

PARTIAL ADJUSTMENT

============================================================== Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 1994 Included observations: 35 after adjusting endpoints ============================================================== Variable Coefficient Std. Error t-Statistic Prob. ============================================================== C -0.390249 0.152989 -2.550839 0.0159 LGDPI 0.313919 0.052510 5.978243 0.0000 LGPRHOUS -0.067547 0.024689 -2.735882 0.0102 LGHOUS(-1) 0.701432 0.045082 15.55895 0.0000 ==============================================================

0.3139 b1 = = 1.05 1 − 0.7014

0.0675 b2 = = 0.23 1 − 0.7014

The long run price elasticity is 0.23, again close to the estimate in the static model. In this example the long-run elasticities are much greater than the short-run ones because the speed of adjustment is slow. 25

PREDICTION

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt

This sequence provides an introduction to the topic of prediction using a regression model. The analysis will be confined to the simple regression model and we will suppose that we have fitted the equation for a sample of observations 1, ..., T (the sample period). 1

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p

Suppose that we know that the value of the x variable at some future time T+p will be xT+p. Then the predicted value of y at that time is found by inserting xT+p into the regression equation. 2

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p

More realistically, we would not know the future value of x but would have to predict it as well. In that case, we should say that the equation gives the predicted value of y, conditional on x being equal to xT+p. 3

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p fT + p = yT + p − yˆ T + p

The difference between the actual value of the dependent variable and the forecast value is the prediction error, fT+p. 4

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p fT + p = yT + p − yˆ T + p E ( fT + p ) = E ( yT + p ) − E ( yˆ T + p ) = E (α + βxT + p + uT + p ) − E (a + bxT + p ) = α + βxT + p + E ( uT + p ) − E (a ) − xT + p E (b ) = α + βxT + p + 0 − α − xT + p β = 0 If the model is correctly specified and if the Gauss-Markov conditions are satisfied, the expected value of the prediction error is 0. We will demonstrate this. 5

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p fT + p = yT + p − yˆ T + p E ( fT + p ) = E ( yT + p ) − E ( yˆ T + p ) = E (α + βxT + p + uT + p ) − E (a + bxT + p ) = α + βxT + p + E ( uT + p ) − E (a ) − xT + p E (b ) = α + βxT + p + 0 − α − xT + p β = 0 We substitute from the true model for the actual value of yT+p and from the regression equation for its predicted value. 6

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p fT + p = yT + p − yˆ T + p E ( fT + p ) = E ( yT + p ) − E ( yˆ T + p ) = E (α + βxT + p + uT + p ) − E (a + bxT + p ) = α + βxT + p + E ( uT + p ) − E (a ) − xT + p E (b ) = α + βxT + p + 0 − α − xT + p β = 0 The first two terms are constants because α and β are constants and we are treating xT+p as known. The last consideration allows us to take xT+p out of the fifth term. 7

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p fT + p = yT + p − yˆ T + p E ( fT + p ) = E ( yT + p ) − E ( yˆ T + p ) = E (α + βxT + p + uT + p ) − E (a + bxT + p ) = α + βxT + p + E ( uT + p ) − E (a ) − xT + p E (b ) = α + βxT + p + 0 − α − xT + p β = 0 We are assuming that the Gauss-Markov conditions are satisfied and hence the third term is 0 and the expected values of a and b are equal to their true values. Thus the expression reduces to 0. 8

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p fT + p = yT + p − yˆ T + p

σ 2f

T+ p

⎧ 1 ( xT + p − x ) 2 ⎫ 2 = ⎨1 + + ⎬σ u ⎩ n nVar( x ) ⎭

The population variance of the forecast error is given by the expression shown.

9

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p fT + p = yT + p − yˆ T + p

σ 2f

T+ p

⎧ 1 ( xT + p − x ) 2 ⎫ 2 = ⎨1 + + ⎬σ u ⎩ n nVar( x ) ⎭

Note that it is sensitive to the distance from the sample mean to the predicted value of x. As is intuitively reasonable, it increases, the further one extrapolates from the sample mean. 10

PREDICTION

yt = α + βxt + ut yˆ t = a + bxt yˆ T + p = a + bxT + p fT + p = yT + p − yˆ T + p

σ 2f

T+ p

⎧ 1 ( xT + p − x ) 2 ⎫ 2 = ⎨1 + + ⎬σ u ⎩ n nVar( x ) ⎭

yˆ T + p − t crit × s.e. < yT + p < yˆ T + p + t crit × s.e. The standard error of the prediction error is calculated using the square root of the expression for the population variance, replacing the population variance of u with the estimate obtained when fitting the model in the sample period. 11

PREDICTION yt confidence interval for yT+p

yˆ T + p = a + bxT + p

yT+p

x

σ 2f

T+ p

xT+p

xt

⎧ 1 ( xT + p − x ) 2 ⎫ 2 = ⎨1 + + ⎬σ u ⎩ n nVar( x ) ⎭

yˆ T + p − t crit × s.e. < yT + p < yˆ T + p + t crit × s.e. Hence we are able to construct a confidence interval for a prediction.

12

PREDICTION yt

upper limit of confidence interval confidence interval for yT+p

yˆ T + p = a + bxT + p

yT+p

lower limit of confidence interval x

σ 2f

T+ p

xT+p

xt

⎧ 1 ( xT + p − x ) 2 ⎫ 2 = ⎨1 + + ⎬σ u ⎩ n nVar( x ) ⎭

yˆ T + p − t crit × s.e. < yT + p < yˆ T + p + t crit × s.e. The confidence interval has been drawn as a function of xT+p. As we noted from the mathematical expression, it becomes wider, the greater the distance from xT+p to the sample mean. 13

PREDICTION ============================================================= Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 1990 Included observations: 32 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 ============================================================= R-squared 0.997730 Mean dependent var 5.936697 Adjusted R-squared 0.997574 S.D. dependent var 0.356814 S.E. of regression 0.017576 Akaike info criter -5.155531 Sum squared resid 0.008958 Schwarz criterion -5.018118 Log likelihood 85.48849 F-statistic 6373.813 Durbin-Watson stat 0.828613 Prob(F-statistic) 0.000000 ============================================================= Here is the result of a logarithmic regression of expenditure on housing services on income and relative price, using the Demand Function data set, taking 1959-1990 as the sample period. We will evaluate how successfully it predicted LGHOUS in the years 1991-1994. 14

PREDICTION

Predicted and Actual Expenditure on Housing Services, 1991-1994 Logarithm Year

^ LGHOUS LGHOUS

Absolute equivalent error

^ HOUS

HOUS

error

1991

6.4374

6.4539

-0.0166

624.8

635.2 -10.4

1992

6.4697

6.4720

-0.0023

645.3

646.8

-1.5

1993

6.4820

6.4846

-0.0026

653.3

655.0

-1.7

1994

6.5073

6.5046

0.0027

670.0

668.2

1.8

The table shows the predicted values, conditional on the actual values of LGDPI and LGPRHOUS, for 1991-1994. 15

PREDICTION

Predicted and Actual Expenditure on Housing Services, 1991-1994 Logarithm Year

^ LGHOUS LGHOUS

Absolute equivalent error

^ HOUS

HOUS

error

1991

6.4374

6.4539

-0.0166

624.8

635.2 -10.4

1992

6.4697

6.4720

-0.0023

645.3

646.8

-1.5

1993

6.4820

6.4846

-0.0026

653.3

655.0

-1.7

1994

6.5073

6.5046

0.0027

670.0

668.2

1.8

yˆ T + p − t crit × s.e. < yT + p < yˆ T + p + t crit × s.e.

In this example, the largest error was for 1991. We will check whether the forecast value was in the 95% confidence interval. 16

PREDICTION

Predicted and Actual Expenditure on Housing Services, 1991-1994 Logarithm Year

^ LGHOUS LGHOUS

Absolute equivalent error

^ HOUS

HOUS

error

1991

6.4374

6.4539

-0.0166

624.8

635.2 -10.4

1992

6.4697

6.4720

-0.0023

645.3

646.8

-1.5

1993

6.4820

6.4846

-0.0026

653.3

655.0

-1.7

1994

6.5073

6.5046

0.0027

670.0

668.2

1.8

yˆ T + p − t crit × s.e. < yT + p < yˆ T + p + t crit × s.e. 6.4374 – 2.045 x 0.0190 < y < 6.4374 + 2.045 x 0.0190 6.3985 < y < 6.4763 The point estimate was 6.4374, the critical value of t at the 5% level with 29 degrees of freedom was 2.045, and the standard error was 0.0190. Hence the actual value did just lie within the confidence interval. 17

PREDICTION

Predicted and Actual Expenditure on Housing Services, 1991-1994 Logarithm Year

^ LGHOUS LGHOUS

Absolute equivalent error

^ HOUS

HOUS

error

1991

6.4374

6.4539

-0.0166

624.8

635.2 -10.4

1992

6.4697

6.4720

-0.0023

645.3

646.8

-1.5

1993

6.4820

6.4846

-0.0026

653.3

655.0

-1.7

1994

6.5073

6.5046

0.0027

670.0

668.2

1.8

yˆ T + p − t crit × s.e. < yT + p < yˆ T + p + t crit × s.e. 6.4374 – 2.045 x 0.0190 < y < 6.4374 + 2.045 x 0.0190 6.3985 < y < 6.4763 How were the standard errors calculated? By defining a dummy variable D1991 equal to 1 in 1991 and 0 in all the other years, and dummy variables D1992, D1993, and D1994 for the other years in the prediction period. 18

PREDICTION

Predicted and Actual Expenditure on Housing Services, 1991-1994 Logarithm Year

^ LGHOUS LGHOUS

Absolute equivalent error

^ HOUS

HOUS

error

1991

6.4374

6.4539

-0.0166

624.8

635.2 -10.4

1992

6.4697

6.4720

-0.0023

645.3

646.8

-1.5

1993

6.4820

6.4846

-0.0026

653.3

655.0

-1.7

1994

6.5073

6.5046

0.0027

670.0

668.2

1.8

yˆ T + p − t crit × s.e. < yT + p < yˆ T + p + t crit × s.e. 6.4374 – 2.045 x 0.0190 < y < 6.4374 + 2.045 x 0.0190 6.3985 < y < 6.4763 Adding these dummy variables to the model, we get the output shown next.

19

PREDICTION ============================================================= Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 D1991 0.016561 0.018991 0.872042 0.3903 D1992 0.002325 0.019033 0.122150 0.9036 D1993 0.002604 0.019109 0.136289 0.8925 D1994 -0.002732 0.019268 -0.141796 0.8882 ============================================================= R-squared 0.998206 Mean dependent var 5.996930 Adjusted R-squared 0.997835 S.D. dependent var 0.377702 S.E. of regression 0.017576 Akaike info criter -5.071925 Sum squared resid 0.008958 Schwarz criterion -4.764019 Log likelihood 98.29465 F-statistic 2689.096 Durbin-Watson stat 0.838021 Prob(F-statistic) 0.000000 =============================================================

20

PREDICTION ============================================================= Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 1994 Included observations: 36 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 D1991 0.016561 0.018991 0.872042 0.3903 D1992 0.002325 0.019033 0.122150 0.9036 D1993 0.002604 0.019109 0.136289 0.8925 D1994 -0.002732 0.019268 -0.141796 0.8882 ============================================================= R-squared 0.998206 Mean dependent var 5.996930 Adjusted R-squared 0.997835 S.D. dependent var 0.377702 S.E. of regression 0.017576 Akaike info criter -5.071925 Sum squared resid 0.008958 Schwarz criterion -4.764019

The estimates of the intercept and the coefficients of LGDPI and LGPRHOUS will be chosen to optimize the fit for the sample period. The coefficients of the dummy variables will then be chosen to obtain a perfect fit for the years 1991-1994. 21

PREDICTION ============================================================= Sample: 1959 1990 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 ============================================================= ============================================================= Sample: 1959 1994 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 D1991 0.016561 0.018991 0.872042 0.3903 D1992 0.002325 0.019033 0.122150 0.9036 D1993 0.002604 0.019109 0.136289 0.8925 D1994 -0.002732 0.019268 -0.141796 0.8882 ============================================================= The estimates of the intercept and the coefficients of LGDPI and LGPRHOUS which optimize the fit for 1959-1990 are of course exactly the same as those where the regression was confined to the sample period. 22

PREDICTION ============================================================= Sample: 1959 1990 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 ============================================================= ============================================================= Sample: 1959 1994 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 D1991 0.016561 0.018991 0.872042 0.3903 D1992 0.002325 0.019033 0.122150 0.9036 D1993 0.002604 0.019109 0.136289 0.8925 D1994 -0.002732 0.019268 -0.141796 0.8882 ============================================================= The observation-specific dummy variables in 1991-1994 guarantee a perfect fit in those years 23

PREDICTION ============================================================= Sample: 1959 1990 ============================================================= Sum squared resid 0.008958

============================================================= Sample: 1959 1994 ============================================================= Sum squared resid 0.008958

You can check this by looking at the residual sum of squares. It is the same as for the original regression. 24

PREDICTION ============================================================= Sample: 1959 1990 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 ============================================================= ============================================================= yˆ 1991 = −1.96 + 1.13 LGDPI − 0.24 LGPRHOUS Sample: 1959 1994 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 D1991 0.016561 0.018991 0.872042 0.3903 D1992 0.002325 0.019033 0.122150 0.9036 D1993 0.002604 0.019109 0.136289 0.8925 y = − 1 . 96 + 1 . 13 LGDPI − 0 . 24 LGPRHOUS + 0.0166 1991 D1994 -0.002732 0.019268 -0.141796 0.8882 ============================================================= The fact that the fitted values for 1991-1994 are identical to the actual values means that the coefficient of D1991 must be equal to minus the prediction error for 1991. 25

PREDICTION ============================================================= Sample: 1959 1990 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 ============================================================= ============================================================= yˆ 1991 = −1.96 + 1.13 LGDPI − 0.24 LGPRHOUS Sample: 1959 1994 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C -1.959273 0.388319 -5.045522 0.0000 LGDPI 1.129984 0.010942 103.2737 0.0000 LGPRHOUS -0.239828 0.075874 -3.160890 0.0037 D1991 0.016561 0.018991 0.872042 0.3903 D1992 0.002325 0.019033 0.122150 0.9036 D1993 0.002604 0.019109 0.136289 0.8925 y = − 1 . 96 + 1 . 13 LGDPI − 0 . 24 LGPRHOUS + 0.0166 1991 D1994 -0.002732 0.019268 -0.141796 0.8882 ============================================================= In turn this means that the standard error of the coefficient of D1991 is the standard error of the prediction error for that year. This is how the 0.0190 used in the confidence interval was derived. 26

PREDICTION

Predicted and Actual Expenditure on Housing Services, 1991-1994 Logarithm Year

^ LGHOUS LGHOUS

Absolute equivalent error

^ HOUS

HOUS

error

1991

6.4374

6.4539

-0.0166

624.8

635.2 -10.4

1992

6.4697

6.4720

-0.0023

645.3

646.8

-1.5

1993

6.4820

6.4846

-0.0026

653.3

655.0

-1.7

1994

6.5073

6.5046

0.0027

670.0

668.2

1.8

yˆ T + p − t crit × s.e. < yT + p < yˆ T + p + t crit × s.e. 6.4374 – 2.045 x 0.0190 < LGHOUS < 6.4374 + 2.045 x 0.0190 6.3985 < LGHOUS < 6.4763 In turn this means that the standard error of the coefficient of D1991 is the standard error of the prediction error for that year. This is how the 0.0190 used in the confidence interval was derived. 27