A researcher investigating the determinants of hamburger sales in a particular restaurant in the US fits the following model:

Sales = + Price + Advert + e

The variables are defined as below.

Sales	Monthly sales revenue of hamburgers sales in thousands of dollars
Price	Price of hamburgers in dollars
Advert	Advertising expenditure in dollars

summarize sales price advert

Variable | Obs Mean Std. Dev. Min Max -------------+--------------------------------------------------------- sales | 75 77.37467 6.488537 62.4 91.2 price | 75 5.6872 .518432 4.83 6.49 advert | 75 1.844 .8316769 .5 3.1

. regress sales price advert

Source | SS df MS Number of obs = 75 -------------+---------------------------------- F(2, 72) = 29.25 Model | 1396.53893 2 698.269465 Prob > F = 0.0000 Residual | 1718.94294 72 23.8742075 R-squared = 0.4483 -------------+---------------------------------- Adj R-squared = 0.4329 Total | 3115.48187 74 42.1011063 Root MSE = 4.8861

------------------------------------------------------------------------------ sales | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- price | -7.907854 1.095993 -7.22 0.000 -10.09268 -5.723032 advert | 1.862584 .6831955 2.73 0.008 .500659 3.22451 _cons | 118.9136 6.351638 18.72 0.000 106.2519 131.5754 ------------------------------------------------------------------------------

. predict u, residuals

. sktest u

Skewness/Kurtosis tests for Normality ------ joint ------ Variable | Obs Pr(Skewness) Pr(Kurtosis) adj chi2(2) Prob>chi2 -------------+--------------------------------------------------------------- u | 75 0.7845 0.4974 0.55 0.7614

. estat vif

Variable | VIF 1/VIF -------------+---------------------- advert | 1.00 0.999305 price | 1.00 0.999305 -------------+---------------------- Mean VIF | 1.00

. estat ovtest

Ramsey RESET test using powers of the fitted values of sales Ho: model has no omitted variables F(3, 69) = 3.15 Prob > F = 0.0303

. estat hettest,rhs

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance Variables: price advert

chi2(2) = 2.80 Prob > chi2 = 0.2470

. estat imtest, white

White's test for Ho: homoskedasticity against Ha: unrestricted heteroskedasticity

chi2(5) = 4.54 Prob > chi2 = 0.4749

Cameron & Trivedi's decomposition of IM-test

--------------------------------------------------- Source | chi2 df p ---------------------+----------------------------- Heteroskedasticity | 4.54 5 0.4749 Skewness | 0.93 2 0.6275 Kurtosis | 0.20 1 0.6576 ---------------------+----------------------------- Total | 5.67 8 0.6846 ---------------------------------------------------

. estat dwatson

Durbin-Watson d-statistic( 3, 75) = 2.183037

. dfuller sales, lags(1) noconstant regress

Augmented Dickey-Fuller test for unit root Number of obs = 73

---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -0.229 -2.611 -1.950 -1.610

------------------------------------------------------------------------------ D.sales | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- sales | L1. | -.0030272 .0131958 -0.23 0.819 -.0293389 .0232845 LD. | -.4656746 .1055288 -4.41 0.000 -.6760931 -.2552562 ------------------------------------------------------------------------------

. dfuller sales, lags(1) regress

Augmented Dickey-Fuller test for unit root Number of obs = 73

---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -7.881 -3.548 -2.912 -2.591 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0000

------------------------------------------------------------------------------ D.sales | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- sales | L1. | -1.365318 .17324 -7.88 0.000 -1.710834 -1.019802 LD. | .2148289 .1159921 1.85 0.068 -.01651 .4461679 | _cons | 105.7835 13.43129 7.88 0.000 78.99565 132.5714 ------------------------------------------------------------------------------

. dfuller sales, lags(1) trend regress

Augmented Dickey-Fuller test for unit root Number of obs = 73

---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -8.361 -4.099 -3.477 -3.166 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0000

------------------------------------------------------------------------------ D.sales | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- sales | L1. | -1.474939 .1764071 -8.36 0.000 -1.826862 -1.123017 LD. | .2697504 .1159291 2.33 0.023 .0384782 .5010226 _trend | .0780826 .0362533 2.15 0.035 .0057593 .1504058 _cons | 111.302 13.34353 8.34 0.000 84.6824 137.9216 ------------------------------------------------------------------------------

Question: Conduct diagnostic tests to check whether the model satisfies the assumptions of OLS and discuss your results. State the null and alternative hypotheses, the test statistic, decision rule and the conclusion for each test.