Yi x0 x0

where, of course, the values of x0 are specified. Note that (C.9.4) gives an unbiased prediction of E(Y- | x0), since E(x'0(3) = x0(3. (Why?)

Variance of Mean Prediction

The formula to estimate the variance of (Y0 | x0) is as follows7:

where a2 is the variance of ui, x0 are the given values of the X variables for which we wish to predict, and (X' X) is the matrix given in (C.3.9). In practice, we replace a2 by its unbiased estimator a2.

We will illustrate mean prediction and its variance in the next section.

7For derivation, see J. Johnston, Econometrics Methods, McGraw-Hill, 3d ed., New York, 1984, pp. 195-196.

942 APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL

Individual Prediction

As pointed out in Chapters 5 and 8, the individual prediction of Y(= Y0) is also given by (C.9.3) or more specifically by (C.9.4). The difference between mean and individual predictions lies in their variances.

Variance of Individual Prediction

The formula for the variance of an individual prediction is as follows8:

where var(Y0 | x0) stands for E[Y0 — Y0 | X]2. In practice we replace a2 by its unbiased estimator a2. We illustrate this formula in the next section.

C.10 SUMMARY OF THE MATRIX APPROACH: AN ILLUSTRATIVE EXAMPLE

Consider the data given in Table C.4. These data pertain to per capita personal consumption expenditure (PPCE) and per capital personal disposable income (PPDI) and time or the trend variable. By including the trend variable in the model, we are trying to find out the relationship of PPCE to PPDI net of the trend variable (which may represent a host of other factors, such as technology, change in tastes, etc.)

For empirical purposes, therefore, the regression model is

where Y = per capita consumption expenditure, X2 = per capita disposable income, and X3 = time. The data required to run the regression (C.10.1) are given in Table C.4.

TABLE C.4 PER CAPITA PERSONAL CONSUMPTION EXPENDITURE (PPCE) AND PER CAPITA PERSONAL DISPOSABLE INCOME (PPDI) IN THE UNITED STATES, 1956-1970, IN 1958 DOLLARS

PPCE, Y

PPDI, X2

Time, X3

PPCE, Y

PPDI, X2

Time, X3

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