1. Regression with Time Series Data
Judge et al Chapter 15 and 16

2. Distributed Lag

3. Polynomial distributed lag

4. Estimating a polynomial distributed lag

5. Geometric Lag
Impact Multiplier: change in yt when xt changes by one unit:
If change in xt is sustained for another period:
Long-run multiplier:

6. The Koyck Transformation

7. Autoregressive distributed lag
ARDL(1,1)
ARDL(p,q)
Represents an infinite distributed lag with weights:
Approximates an infinite lag of any shape when p and q are large.

8. Stationarity
The usual properties of the least squares estimator in a regression using time series data depend on the assumption that the variables involved are stationary stochastic processes.
A series is stationary if its mean and variance are constant over time, and the covariance between two values depends only on the length of time separating the two values

9. Stationary Processes

10. Non-stationary processes

11. Non-stationary processes with drift

12. Summary of time series processes
Random walk
Random walk with drift
Deterministic trend

13. Trends Stochastic trend Deterministic trend Random walk
Series has a unit root
Series is integrated I(1)
Can be made stationary only by first differencing
Deterministic trend
Series can be made stationary either by first differencing or by subtracting a deterministic trend.

14. Real data

15. Spurious correlation
Two random walks we observed earlier.

16. Spurious regression R2 0.7495 D-W 0.0305 Variable DF B Value Std Error
T ratio
Approx prob
Intercept 1
0.5429
26.162
0.0001
RW2 R
D-W

17. Checking/testing for stationarity
Correlogram
Shows partial correlation observations at increasing intervals.
If stationary these die away.
Box-Pierce
Ljung-Box
Unit root tests

18. Unit root test If || < 1, then the AR(1) process is stationary.
We can test for nonstationarity by testing the null hypothesis that  = 1 against the alternative that , or simply  < 1.

19. Dickey Fuller Tests Allow for a number of possible models
Drift
Deterministic trend
Account for serial correlation
Drift
Drift against deterministic trend
Adjusting for serial correlation (ADF)

20. Critical values Table 16.4 Critical Values for the Dickey-Fuller Test
Model 1% 5%
10%
2.56
1.94
1.62
3.43
2.86
2.57
3.96
3.41
3.13
Standard critical values
2.33
1.65
1.28

21. Example of a Dickey Fuller Test

22. Cointegration
In general non-stationary variables should not be used in regression.
In general a linear combination of I(1) series, eg: is I(1).
If et is I(0) xt and yt are cointegrated and the regression is not spurious
et can be interpreted as the error in a long-run equilibrium.

23. Example of a cointegration test
Model 1% 5%
10%
3.90
3.34
3.04