Multiple regression model

Question

I have a multiple regression equation which as four quarters (maybe called them as parameters)

quartt = beta_0 + beta_1*t + beta_2*(t^2) + beta_3*Q_1 + beta_4*Q_2 + beta5*Q_3 + error_t

R result:

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 276.6363 35.0485 7.893 3.43e-09 ***
t -7.4582 3.3960 -2.196 0.034999 *
I(t^2) 0.3012 0.0803 3.751 0.000657 ***
as.factor(Q)1 65.7707 27.1592 2.422 0.020933 *
as.factor(Q)2 -37.8701 27.0958 -1.398 0.171275
as.factor(Q)3 -127.6113 27.0574 -4.716 3.99e-05 ***
---
Residual standard error: 60.47 on 34 degrees of freedom
Multiple R-squared: 0.7443, Adjusted R-squared: 0.7067
F-statistic: 19.8 on 5 and 34 DF, p-value: 3.303e-09

Question:

1. E(quartt|Q3) = beta_0 +beta_1*t + beta_2*(t^2) + beta_5*(Q_3) Is it correct?

2. Do the independent variables, t, t^2, and Q in the model statistically significant?

I think they do since the P-value of F-test is small enough to reject the null by using alpha = 0.05. However, can I say even though the p-value of Q2 is not less than alpha, since Q1 and Q3 are less than alpha, we can say that Q is statistically significant simultaneously?

Thank you!!

Answer 1

Your first interpretation is correct, the expected value of the dependent value conditional on the third quarter (and on the remaining regressor variables) is defined as you did.

Given a significance level $\alpha=0.05$, all the variables except the second quarter are individually significant since the p-values related to the $t$-test statistics are lower than $0.05$.

The $F$-test statistic that you get is related to the null that all the regressor variables (except the intercept) are zero.

In this context, it is more relevant to test for the significance of the seasonal dummy variables, that is, test the null $\beta_3=\beta_4=\beta_5=0$. The test statistic for this hypothesis is not reported in the output that you show.

You can test this hypothesis upon the sum of squared residuals from the unrestricted and the restricted models. The former is the model that you have already fitted, while the latter is the model where the null hypothesis is imposed, that is, quartt = beta_0 + beta_1*t + beta_2*(t^2) + error_t. The R function anova can be used to test this hypothesis.

tsq <- (1:length(quartt))^2
unrestricted.model <- lm(quartt ~ tsq + Q1 + Q2 + Q3)
restricted.model <- lm(quartt ~ tsq)
anova(unrestricted.model, restricted.model)