Springer Texts in Business and Economics
Numerical Example
Table 3.1 gives the annual consumption of 10 households each selected randomly from a group of households with a fixed personal disposable income. Both income and consumption are measured in $10,000, so that the first household earns $50, 000 and consumes $46,000 annually. It is worthwhile doing the computations necessary to obtain the least squares regression estimates of consumption on income in this simple case and to compare them with those obtained from a regression package. In order to do this, we first compute Y = 6.5 and X = 7.5 and form two new columns of data made up of yi = Yi — Y and xi = Xi — X. To get /3OLS we need £™=1 ХіУі, so we multiply these last two columns by each other and sum to get 42.5. The denominator of eOLS is given by £n=i x2. This is why we square the xi column to get x2 and sum to obtain 52.5. Our estimate of /3OLS = 42.5/52.5 = 0.8095 which is the estimated marginal propensity to consume. This is the extra consumption brought about by an extra dollar of disposable income.
3OLS = Y — 3OLS X = 6.5 — (0.8095)(7.5) = 0.4286
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This is the estimated consumption at zero personal disposable income. The fitted values or predicted values from this regression are computed from Y = 33OLS + /3OLSXi = 0.4286 + 0.8095Xi and are given in Table 3.1. Note that the mean of Y3i is equal to the mean of Yi confirming one of the numerical properties of least squares. The residuals are computed from ei = Yi — 3 and they satisfy £™=1 ei = 0. It is left to the reader to verify that £™=1 eiXi = 0. The residual sum of squares is obtained by squaring the column of residuals and summing it. This gives us £П=і ef = 2.495238. This means that s2 = £™=1 e2/(n — 2) = 0.311905. Its square root is given by s = 0.558. This is known as the standard error of the regression. In this case, the estimated var(POLS) is s2/Y^i=1 xf = 0.311905/52.5 = 0.005941 and the estimated
Taking the square root of these estimated variances, we get the estimated standard errors of 33OLS and eOLS given by se(3OLS) = 0.60446 and se(/3OLS) = 0.077078.
Since the disturbances are normal, the OLS estimators are also the maximum likelihood estimators, and are normally distributed themselves. For the null hypothesis H0g; в = 0; the observed t-statistic is
tabs = (Pols — 0)./se(pOLS) = 0.809524/0.077078 = 10.50
and this is highly significant, since Pr[|fg| > 10.5] < 0.0001. This probability can be obtained using most regression packages. It is also known as the p-value or probability value. It shows that this t-value is highly unlikely and we reject Hg that в = 0. Similarly, the null hypothesis Hb ; a = 0, gives an observed t-statistic of tabs = (3OLS — 0)/Se(3OLS) = 0.428571/0.604462 = 0.709, which is not significant, since its p-value is Pr[|tg| > 0.709] < 0.498. Hence, we do not reject the null hypothesis H0 that a = 0.
The total sum of squares is £™=1 y2 = £i=1(Yi — F)2 which can be obtained by squaring the yi column in Table 3.1 and summing. This yields £™=1 y2 = 36.9. Also, the regression sum of
squares = £i=132 = £™=1(Y — F)2 which can be obtained by subtracting Y = Y = 6.5 from the Yi column, squaring that column and summing. This yields 34.404762. This could have also been obtained as
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En=i УІ = воьзТ;П=і X = (0.809524)2(52.5) = 34.404762.
R2 = 4 = £ n=1 n=1 У2 = 34.404762/36.9 = 0.9324.
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This means that personal disposable income explains 93.24% of the variation in consumption. A plot of the actual, predicted and residual values versus time is given in Figure 3.8. This was done using EViews.
