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Autocorrelation Lecture 20 Lecture 20

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**Today’s plan Definition and implications**

How to test for first order autocorrelation Note: we’ll only be taking a detailed look at 1st order autocorrelation, but higher orders exist e.g. quarterly data is likely to have 4th order autocorrelation How to correct for first-order autocorrelation and how to estimate allowing for autocorrelation Again we’ll use the Phillips curve as an example Lecture 20

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**Definitions and implications**

Autocorrelation is a time-series phenomenon 1st-order autocorrelation implies that neighboring observations are correlated the observations aren’t independent draws from the sample Lecture 20

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**Definitions and implications (2)**

In terms of the Gauss-Markov (or BLUE) theorem: The model is still linear and unbiased if autocorrelation exists: Lecture 20

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**Definitions and implications (3)**

Autocorrelation will affect the variance: if s = t, then we would have: but if s t, and Y observations are not independent, we have nonzero covariance terms: Lecture 20

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**Definitions and implications (4)**

Think of a numerical example to demonstrate this assuming t = 3: if we wanted to estimate : We want to consider the efficiency, or the variance of If BLUE: all covariance terms are zero. If covariance terms are nonzero: we no longer have minimum variance minimum variance is defined as Lecture 20

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**Summary of implications**

1) Estimates are linear and unbiased 2) Estimates are not efficient. We no longer have minimum variance 3) Estimated variances are biased either positively or negatively 4) Unreliable t and F test results 5) Computed variances and standard errors for predictions are biased Main idea: autocorrelation affects the efficiency of estimators Lecture 20

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**How does autocorrelation occur?**

Autocorrelation occurs through one of the following avenues: 1) Intertia in economic series through construction With regards to unemployment this was called hysteresis: this means that certain sections of society who are prone to unemployment 2) Incorrect model specification There might be missing variables or we might have transformed the model to create correlation across variables Lecture 20

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**How does autocorrelation occur?**

3) Cobweb phenomenon agents respond to information with lags to this is usually related to agricultural markets 4) Data manipulation example: constructing annual information based on quarterly data Lecture 20

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Graphical results With no autocorrelation in the error term: we would expect all errors to be randomly dispersed around zero within reasonable boundaries Simply graphing the estimated errors against time indicates the possibility of autocorrelation: we look for patterns of errors over time patterns can be positive, negative, or zero Graph error vs. time, we have positive correlation in the error term errors from one time period to the next tend to move in 1 direction, with a positive slope Lecture 20

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**Phillips Curve L_20.xls : Phillips Curve data**

Can calculate predicted wage inflation using the observed unemployment rate and the estimated regression coefficients Can then calculate the estimated error of the regression equation Can also calculate the error lagged one time period Lecture 20

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**Durbin-Watson statistic**

We will use the Durbin-Watson statistic to test for autocorrelation This is computed by looking over T-1 observations where t = 1, …T Lecture 20

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**Durbin-Watson statistic (2)**

The assumptions behind the Durbin-Watson statistic are: 1) You must include an intercept in the regression 2) Values of X are independent and fixed 3) Disturbances, or errors, are generated by: this says that errors in this time period are correlated with errors in the last time period and some random error vt is the coefficient of autocorrelation and is bounded -1 1 can be calculated as Lecture 20

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How to estimate This estimation matters because it will be used in the model correction can be estimated by this equation: Once we have the Durbin-Watson statistic d, you can obtain an estimate for Lecture 20

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**How to estimate (2) How the test works:**

The values for d range between 0 and 4 with 2 as the midpoint using : Lecture 20

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**How to estimate (3) We can represent this in the following figure: 2**

4 H1 dL dU H0: =0 Reject null Cannot reject null 4-dU 4-dL indeterminate dL represents the D-W upper bound dU represents the D-W lower bound The mirror image of dL and dU are 4- dL and 4-dU Lecture 20

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Procedure Table on the second handout for today is the Durbin-Watson statistical table and an additional table for this analysis 1) Run model: 2) Compute: 3) Compute d statistic 4) Find dL and dU from the tables K’ is the number of parameters minus the constant and T is the number of observations 5) Test to see if autocorrelation is present Lecture 20

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**Example (2) Returning to L_20.xls 2 4 H1 dL dU H0: =0 Reject null**

H1 dL dU H0: =0 Reject null Accept null 4-dU 4-dL 0.331 1.475 Lecture 20

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**Generalized least squares**

What can we do about autocorrelation? Recall that our model is: Yt = a + bXt + et (1) We also know: et = et-1 + vt We will have to estimate the model using generalized least squares (GLS) Lecture 20

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**Generalized least squares (2)**

Let’s take our model and lag it by one time period: Yt-1 = a + bXt-1 + et-1 Multiplying by : Yt-1 = a + bXt-1 + et (2) Subtracting our (2) from (1), we get Yt - Yt-1 = a(1-) + b(Xt-1 - Xt-1) + vt where vt = et - et-1 Lecture 20

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**Generalized least squares (3)**

Now we need an estimate of : we can transform the variables such that: where: Estimating equation (3) allows us to estimate without first-order autocorrelation Lecture 20

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**Estimating There are several approaches**

One way is by using a short cut: thinking back to the Durbin-Watson statistic, we can rewrite the expression for d as: Lecture 20

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**Estimating (2) Collecting like terms, we have:**

Solving for , we can get an estimate in terms of d: Since earlier we defined as: we can use this to get a more precise estimate There are three or four other methods in the text Lecture 20

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