For this regression, we use the following variables: insurance expenditure (minsur), health expenditure (mxhea), and total monthly income (totminc).
regress mxinsur mxhea totminc
Source | SS df MS Number of obs = 959 ---------+------------------------------ F( 2, 956) = 447.20 Model | 19264987.7 2 9632493.86 Prob > F = 0.0000 Residual | 20591887.5 956 21539.6313 R-squared = 0.4834 ---------+------------------------------ Adj R-squared = 0.4823 Total | 39856875.2 958 41604.2539 Root MSE = 146.76 ------------------------------------------------------------------------------ mxinsur | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- mxhea | .9232861 .0699945 13.191 0.000 .7859255 1.060647 totminc | .0248329 .0010183 24.386 0.000 .0228345 .0268312 _cons | 2.760581 5.220382 0.529 0.597 -7.484149 13.00531 ------------------------------------------------------------------------------
The F statistic test the following hypothesis for this regression:
H0: beta1 = beta2 = 0
H1: beta1 does not equal 0 or beta2 does not equal 0 or neither beta1 nor beta2 equals 0,
where beta1 is the coefficient for mxhea in the population and beta2 is the coefficient for totminc in the population. The null hypothesis (H0) states, in words, that both coefficients equal 0, i.e. that no relationship exists between health care expenditure and insurance expenditure nor total monthly income and insurance expenditure in the population. The F statistic of 447.2 means that the proportion of explained to unexplained variance from our sample is 447 standard deviations above the expected proportion of explained to unexplained variance under the assumption that the null hypothesis is true (which, if we are to believe that no relationships exist, would be 0). If this null hypothesis is true for the population, our model has certainly achieved an extraordinary result; so extraordinary in fact, that we should doubt that this null hypothesis is true in the population. Thus either health insurance expenditure or total monthly income or both have an effect on insurance expenditure in the population. The fact that we can safely reject this null hypothesis is reflected in the value for "Prob > F" of 0.0000. This figure tells us that we can reject the null hypothesis with a less that 0.01% chance of being wrong.
The t statistic for totminc, for example, tests the following null hypothesis:
H0: beta2 = 0
H1: beta2 does not equal 0,
where beta2 is again the multiple regression slope coefficient for insurance expenditure on total monthly income in the population. The t statistic simply tests that beta2 is significantly different from 0 using sample data. Under the assumption that this null hypothesis is true, the sampling distribution for beta2 has an expected value of 0 (if no relationship exists we would expect total monthly income to explain none of the variation in insurance expenditure) and a standard error of 0.0010183. Dividing the coefficient that we get from the sample, 0.0248, by the standard error yields our t statistic of 24.38, meaning that we our sample coefficient is over 24 standard deviations above the value of 0 (which we would expect if the null hypothesis is true). You might recall that for large samples (n > 30), t statistics greater than 2 are usually grounds for rejection of the null hypothesis. Hence in this case, we can safely reject the null hypothesis. The value of 0.000 for "P > | t |" indicates that we can reject the notion that total monthly income does not influence insurance expenditure with a less than 0.01% chance of being wrong. The interpretation for the coefficient for health care expenditure (or any variable employed as a regressor) is completely analogous.