# Thread: Independent and Serial Correlation

1. ## Independent and Serial Correlation

I have a set of 76 annual occurrences of x and y. When I run a linear regression, y is significantly correlated to x. The Durbin-Watson test shows y is also serially correlated.

My goal is to predict y from a certain future x and to determine the prediction interval.

I believe the proper point estimate equation is: Yt = (Xt*B) + (p*Yt-1) – (p*Xt-1*B) + Vt. Is this correct?

Xt, Yt-1, and Xt-1 are known. What are B, p, and Vt and how can I calculate them for a particular Yt prediction?

How do I calculate the prediction interval for Yt?

If not too complex, how does equation change if serial correlation is second order?

2. ## Re: Independent and Serial Correlation

Y being correlated to X is not the same thing as autocorrelation. The former shows that X predicts Y, the later that the residuals are correlated with each other I believe, because of the nature of time series data commonly.

Unless you are using X and Y differently than commonly done. I am not certain of the logic of predicting a present Y from a future X. One of the requirements logically (not statistically) for X to predict Y is that X must occur before Y.

I am guessing you are using notation different than I have encountered in the past.

3. ## Re: Independent and Serial Correlation

I inaccurately stated my goal. It is to predict a future y from a certain future x. For example, my correlation equation predicts y from a certain x. Ignoring y's serial correlation, I could use that equation to predict the next annual y from an assumed next annual x. Without serial correlation, I know how to calculate y's prediction interval. How do I calculate y's prediction interval if y is also serially correlated?

4. ## Re: Independent and Serial Correlation

Have a look at Newey-West standard errors also named heteroskedasticity-and-autocorrelation concistent (HAC) standard errors....

5. ## Re: Independent and Serial Correlation

Hi,

If I understand correctly, you ran OLS and found out that Y is correlated with X, but that autocorrelation is present which you want to include in your model. There are multiple ways I believe, but an ARIMA(X) model should address this quite simple.

In SPSS, go to forecasting-->create models. In the covariates section you place your X variable. Under the dependent variable...(you know..). As method you select the ARIMA model. Go to criteria, there you will find all zero's, which you have to change later on. Keep it zero's in the first place. In the 'statistics' tab you tick the boxes 'r-square', 'display forecasts' and parameter estimates. Then, in the PLOT tab, under individual models, you select the ACF and PACF function.

Press the OK button and show us the ACF and PACF plot. They should provide information.

6. ## Re: Independent and Serial Correlation

Originally Posted by JesperHP
Have a look at Newey-West standard errors also named heteroskedasticity-and-autocorrelation concistent (HAC) standard errors....
As expected, Newey-West produces larger standard errors for b1 and b2 but those standard errors do not impact a particular predicted y's confidence interval. The formula for confidence interval does not use those standard errors.

7. ## Re: Independent and Serial Correlation

I meant the larger standard errors do not affect a particular y's prediction interval.
Originally Posted by cisaak
As expected, Newey-West produces larger standard errors for b1 and b2 but those standard errors do not impact a particular predicted y's confidence interval. The formula for confidence interval does not use those standard errors.

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