Question

1. Suppose we estimate the regression $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + u$, using 282 observations. We then conduct several hypothesis tests with significance level $\alpha = 0.05$: F-test for goodness of fit, two Breusch-Pagan tests, and two White tests (using the predicted dependent variable in a cubic specification). Complete the table below (critical values, test statistics, p-values, and conclusions of each test). F-test for goodness of fit Breusch-Pagan ($F$) Breusch-Pagan ($LM$) White ($F$) White ($LM$) c.v. t.s. p 3.8694 2.1 8.05 0.135 0.0421 conclusion 2. In the model $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + u$, we estimate $\hat{\beta}_1 = 4$ and $\hat{\beta}_2 = -2$. Suppose that $var(u_i)$ is decreasing with $x_1$. Sketch what we would expect the plot of the residuals against $\hat{y}$ to look like. $\hat{u}^2$ 3. Suppose we estimate the regression $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + u$. We find $\hat{\beta}_1 = 2.5$, $\hat{\beta}_2 = -0.73$, SST = 10,000, and $R^2 = 0.45$. We then compute $\sum_{i=2}^n (\hat{u}_i - \hat{u}_{i-1})^2 = 18,425$. The Durbin-Watson critical values for $\alpha = 0.05$ are $d_L = 0.77$ and $d_U = 0.82$. What can we conclude about autocorrelation in this regression? Justify.

          1. Suppose we estimate the regression $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + u$, using 282 observations. We then conduct
several hypothesis tests with significance level $\alpha = 0.05$: F-test for goodness of fit, two Breusch-Pagan
tests, and two White tests (using the predicted dependent variable in a cubic specification). Complete
the table below (critical values, test statistics, p-values, and conclusions of each test).
F-test for goodness of fit
Breusch-Pagan ($F$)
Breusch-Pagan ($LM$)
White ($F$)
White ($LM$)
c.v.
t.s.
p
3.8694
2.1
8.05
0.135
0.0421
conclusion
2. In the model $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + u$, we estimate $\hat{\beta}_1 = 4$ and $\hat{\beta}_2 = -2$. Suppose that $var(u_i)$ is
decreasing with $x_1$. Sketch what we would expect the plot of the residuals against $\hat{y}$ to look like.
$\hat{u}^2$
3. Suppose we estimate the regression $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + u$. We find $\hat{\beta}_1 = 2.5$, $\hat{\beta}_2 = -0.73$,
SST = 10,000, and $R^2 = 0.45$. We then compute $\sum_{i=2}^n (\hat{u}_i - \hat{u}_{i-1})^2 = 18,425$. The Durbin-Watson
critical values for $\alpha = 0.05$ are $d_L = 0.77$ and $d_U = 0.82$. What can we conclude about autocorrelation
in this regression? Justify.

1. Suppose we estimate the regression y = β0 + β1x1 + β2x2 + u, using 282 observations. We then conduct
several hypothesis tests with significance level α = 0.05: F-test for goodness of fit, two Breusch-Pagan
tests, and two White tests (using the predicted dependent variable in a cubic specification). Complete
the table below (critical values, test statistics, p-values, and conclusions of each test).
F-test for goodness of fit
Breusch-Pagan (F)
Breusch-Pagan (LM)
White (F)
White (LM)
c.v.
t.s.
p
3.8694
2.1
8.05
0.135
0.0421
conclusion
2. In the model y = β0 + β1x1 + β2x2 + u, we estimate β̂1 = 4 and β̂2 = -2. Suppose that var(ui) is
decreasing with x1. Sketch what we would expect the plot of the residuals against ŷ to look like.
û^2
3. Suppose we estimate the regression y = β0 + β1x1 + β2x2 + u. We find β̂1 = 2.5, β̂2 = -0.73,
SST = 10,000, and R^2 = 0.45. We then compute ∑i=2^n (ûi - ûi-1)^2 = 18,425. The Durbin-Watson
critical values for α = 0.05 are dL = 0.77 and dU = 0.82. What can we conclude about autocorrelation
in this regression? Justify.

Added by Jennifer B.

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