The Autocorrelation Function and AR(1), AR(2) Models
The Autocorrelation Function and AR(1), AR(2) Models
Al Nosedal University of Toronto
January 29, 2019
Al Nosedal University of Toronto
The Autocorrelation Function and AR(1), AR(2) Models January 29, 2019
1 / 82
Motivation
Autocorrelation, or serial correlation, occurs in data when the error terms of a regression forecasting model are correlated. When autocorrelation occurs in a regression analysis, several possible problems might arise. First, the estimates of the regression coefficients no longer have the minimum variance property and may be inefficient. Second, the variance of the error terms may be greatly underestimated by the mean square error value. Third, the true standard deviation of the estimated regression coefficient may be seriously underestimated. Fourth, the confidence intervals and tests using t and F distributions are no longer strictly applicable.
Al Nosedal University of Toronto
The Autocorrelation Function and AR(1), AR(2) Models January 29, 2019
2 / 82
Motivation (cont.)
First-order autocorrelation results from correlation between the error terms of adjacent time periods (as opposed to two or more previous periods). If first-order autocorrelation is present, the error for one time period et, is a function of the error of the previous time period, et-1, as follows:
et = et-1 + wt The first-order autocorrelation coefficient, , measures the correlation between the error terms; wt is a Normally distributed independent error term. If the value of is 0, et = wt, which means there is no autocorrelation and et is just a random, independent error term.
Al Nosedal University of Toronto
The Autocorrelation Function and AR(1), AR(2) Models January 29, 2019
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Durbin-Watson Test
One way to test to determine whether autocorrelation is present in a time-series regression analysis is by using the Durbin-Watson test for autocorrelation.
D=
n t =2
(et
-
et-1)2
n t =1
et2
where n = the number of observations.
Al Nosedal University of Toronto
The Autocorrelation Function and AR(1), AR(2) Models January 29, 2019
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Durbin-Watson Test (cont.)
The range of values of D is 0 D 4 where small values of D (D < 2) indicate a positive first-order autocorrelation and large values of D (D > 2) imply a negative first-order autocorrelation. Positive first-order autocorrelation is a common occurrence in business and economic time series. The null hypothesis for this test is that there is no autocorrelation. A one-tailed test is used:
H0 : = 0 vs Ha : > 0
In the Durbin-Watson test, D is the observed value of the Durbin-Watson statistic using the residuals from the regression analysis. Our Tables are designed to test for positive first-order autocorrelation by providing values of dL and dU for a variety of values of n and k and for = 0.01 and = 0.05.
Al Nosedal University of Toronto
The Autocorrelation Function and AR(1), AR(2) Models January 29, 2019
5 / 82
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