1. Introduction to Econometrics, 5th edition
Type author name/s here
Dougherty
Introduction to Econometrics, 5th edition
Chapter heading
Chapter 11: Models Using Time Series Data
© Christopher Dougherty, All rights reserved.

2. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
Although they are biased in finite samples if Part (2) of Assumption C.7 is violated, OLS estimators are consistent if Part (1) is valid. We will demonstrate this for the univariate process shown, the simplest of its type. A necessary condition is |b2| < 1. 1

3. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
Provided that this condition is satisfied, the process has mean zero. To demonstrate this, first express Yt as a linear combination of current and lagged values of ut. The weights decline because | b2 | < 1 and tend to zero. 2

4. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
We then take expectations and all the terms on the right side are zero. 3

5. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
The process has variance su2 / (1 – b22). We will take this on trust for the time being. It is demonstrated, with clarifications concerning what is meant by the mean and variance of a time series process, at the beginning of Chapter 13. 4

6. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
The OLS estimator of b2 has been decomposed in the usual way. 5

7. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
The OLS estimator is consistent if the plim of the error term is zero. 6

8. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
if A and B have probability limits
and plim B is not 0.
Since the error term is a ratio, we would like to make use of the plim quotient rule. However, we can do this only if the numerator and denominator have plims, and if the plim of the denominator is not zero. 7

9. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
if A and B have probability limits
and plim B is not 0.
As they stand, the numerator and the denominator of the error term do not converge to limits as the same size becomes large. 8

10. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
The first step is to divide both by the number of observations, in this case T – 1. (We cannot use the first observation because Yt–1 is not defined.) 9

11. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
Before we go any further, it is helpful to establish the relationship shown. 10

12. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
We will use this result rearranged as shown. 11

13. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
We know that the plim of the original expression is cov(X,Y), provided that certain assumptions are valid. 12

14. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
Hence we obtain the plim of the rearranged expression. 13

15. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
By considering the special case where Y = X, we also obtain this result. 14

16. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
We now apply these general results to the present analysis. cov(Yt–1, ut) is zero because ut is generated randomly after Yt–1 has been determined. We have shown that the mean of the process for Y is zero, as is the population mean of the disturbance term, by assumption. 15

17. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
The variance is as shown. (We are taking this on trust. For a proof and a discussion of what is meant by the mean and variance of a process, see the first section of Chapter 13.) 16

18. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
if A and B have probability limits
and plim B is not 0.
We have demonstrated that both the numerator and the denominator of the error term have plims, and hence that we are entitled to use the plim quotient rule. The plim of the denominator is not zero because variances are positive. 17

19. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES
Thus the plim of the error term is zero. Hence the OLS estimator is consistent. 18

20. Copyright Christopher Dougherty 2016.
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