1 Another useful model is autoregressive model. Frequently, we find that the values of a series of financial data at particular points in time are highly correlated with the value which precede and succeed them. Autoregressive models

2 Models with lagged variable Dependent variable is a function of itself at the previous moment of period or time. The creation of an autoregressive model generates a new predictor variable by using the Y variable lagged 1 or more periods.

3 The most often seen form of the equation is a linear form: where: y t – the dependent variable values at the moment t, y t-i (i = 1, 2,..., p) – the dependent variable values at the moment t-i, bo, bi (i=1,..., p) – regression coefficient, p – autoregression rank, e t – disturbance term.

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5 A first-order autoregressive model is concerned with only the correlation between consecutive values in a series. A second-order autoregressive model considers the effect of relationship between consecutive values in a series as well as the correlation between values two periods apart.

6 The selection of an appropriate autoregressive model is not an easy task. Once a model is selected and OLS method is used to obtain estimates of the parameters, the next step would be to eliminate those parameters which do not contribute significantly.

7 (The highest-order parameter does not contribute to the prediction of Yt) (The highest-order parameter is significantly meaningful)

8 using an alpha level of significance, the decision rule is to reject H 0 if or if and not to reject H 0 if

9 Some helpful information:

10 If the null hypothesis is NOT rejected we may conclude that the selected model contains too many estimated parameters. The highest-order term then be deleted an a new autoregressive model would be obtained through least- squares regression. A test of the hypothesis that the “new” highest-order term is 0 would then be repeated.

11 This testing and modeling procedure continues until we reject H 0. When this occurs, we know that our highest-order parameter is significant and we are ready to use this model.

12 Example 1

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18 We have to estimate the parameters of the first-order autoregressive model: and then check if Beta1 is statistically significant.

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20 Example 2

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24 Autogregressive Modeling  Used for Forecasting  Takes Advantage of Autocorrelation  1st order - correlation between consecutive values  2nd order - correlation between values 2 periods apart  Autoregressive Model for pth order: Random Error

25 Autoregressive Modeling Steps  1. Choose p:  2. Form a series of “lag predictor” variables Y i-1, Y i-2, … Y i-p  3. Use Excel to run regression model using all p variables  4. Test significance of B p  If null hypothesis rejected, this model is selected  If null hypothesis not rejected, decrease p by 1 and repeat your calculations