Outline I.  Trends: Linear vs. Exponential II  Using Trending Variables in Regression III. Detrending Interpretation IV.  R‐Squared with trending y V.  Seasonality

12.  Trends and Seasonality Time Series Analysis Read Wooldridge, (2013) Chapter 10.5

I. Trends 12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat

• An economic time series have a common  tendency of growing overtime. Thus, some time  series contain time trend.

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• Models capturing trending behavior:

•

{et } is an independently, identically distributed (i.i.d.) sequence of unobservable.

II. Using III. Detrending IV. R-Squared V. Seasonality

What is 1? Let et = 0

yt = yt – yt-1 = 1 • •

12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat

Write a time series {yt } as

yt = 0 + 1t + et

– (1) Linear time trend – (2) exponential trend 

I. Trends

2

Linear time trend

I.  Trends

I. Trends

II. Using III. Detrending IV. R-Squared V. Seasonality

12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat

3

If 1 > 0, then yt has an upward trend (growing overtime) If 1 < 0, then yt has a downward trend. (shrinking overtime)

I. Trends I. Trends

II. Using III. Detrending IV. R-Squared V. Seasonality 12. Trends and Seasonality . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat

4

Linear time trend

Usefulness of a linear time trend:  Example: Linear time trend Let baht$ be the exchange rate (bath/$) $ = 16.0 + .4143t s.e. (1.37)  (.054) t‐stat [11.64] [7.61]

• yt = 0 + 1t + et • E(yt) is linear in t.  yt = 0 + 1t + et E(yt) = 0 + 1t

n=43 (1960‐2002), R2=.585749, R2‐bar=.575645

• If 1 > 0, then yt has an upward trend (growing overtime) • If 1  0; 2  0  3