# 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:

$$Y_t = X_tB + pY_{t-1} – pX_{t-1}B + V_t.$$

Is this correct?

$X_t$, $Y_{t-1}$, and $X_{t-1}$ are known. What are $B$, $p$, and $V_t$ and how can I calculate them for a particular $Y_t$ prediction?

How do I calculate the prediction interval for $Y_t$?

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

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Welcome to our site! Please check that the $\TeX$ markup I applied did not change the meaning of your equations. –  whuber Aug 15 '12 at 18:56
I don't understand what B represents. I also don't know why you would use the same p for the coefficient of both Y$_t$$_-$$_1$ and X$_t$$_-$$_1$ . Is V$_t$ suppose to be your random noise term? –  Michael Chernick Aug 15 '12 at 19:35
@MichaelChernick, $B$ is the backshift operator. $X_tB=X_{t-1}$. –  Max Oct 4 '12 at 16:21
@Max I am very familiar with the backshift operator as used by Box and Jenkins in their book. But standard notation would be BX$_t$=X$_t$$_-$$_1$ and not with b coming after the variable it is operated on. So I thought it might be something else like a constant or a time independent variable. It is just not standard notation. Also if you are going to apply the operator to the Xs why not apply it to the Ys as well? –  Michael Chernick Oct 4 '12 at 16:35
@MichaelChernick, That is a very good point. And suddenly I am not so sure of its meaning anymore. –  Max Oct 4 '12 at 16:44
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A model of the form Y$_t$ =a$_1$ X$_t$ + c$_1$ Y$_t$$_-$$_1$ + a$_2$ X$_t$$_-$$_1$ +V$_t$ would be appropriate to model a first order autoregressive serial correlation with a covariate that is related to Y$_t$ at times t and t-1.

You can fit the model by minimizing the sum of squared residuals. You can predict Y$_t$ using the right hand side of the above equation (V$_t$ excluded) with the fitted parameter estimates plugged in.

Actually when Y$_t$$_-$$_1$ is used in the model the serial correlation is non-zero at every lag. It will be p$^i$ at lag i. However if what you meant is that you want to incorporate a dependence on Y$_t$$_-$$_2$ then just add a term of the form c$_2$ Y$_t$$_-$$_2$.

The quantity p appears to be a model parameter that you estimate from the time series data. The V$_t$ is probably a random noise component. I am not sure what B represents here. It could be the backshift operator applied to X$_t$ and X$_t$$_-$$_1$ but the usual notation is to place the operator in frontt og the variable it is operating on rather than behind.

Since you introduced the equation to us it seems that you should know what the terms mean. We have no context other than what you gave us.

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I am lost. How do I calculate a1, c1, and a2? Once I predict Yt, how do I calculate the prediction interval? –  user13332 Sep 4 '12 at 15:10