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1 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 | 2 | < 1. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

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2 Provided that this condition is satisfied, the process has mean zero. To demonstrate this, first express Y t as a linear combination of current and lagged values of u t. The weights decline because | b 2 | < 1 and tend to zero.

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

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4 CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES The process has variance u 2 / (1 – 2 2 ). 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.

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5 The OLS estimator of 2 has been decomposed in the usual way. CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES

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

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7 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. if A and B have probability limits and plim B is not 0.

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8 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.

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9 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 Y t–1 is not defined.)

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

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

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12 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.

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

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

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15 CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES We now apply these general results to the present analysis. cov(Y t–1, u t ) is zero because u t is generated randomly after Y t–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.

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16 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).

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17 CONSISTENCY OF OLS ESTIMATOR FOR A UNIVARIATE TIME SERIES 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. if A and B have probability limits and plim B is not 0.

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

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Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 11.5 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics

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