Question
A researcher investigating the determinants of hamburger sales in a particular restaurant in the US fits the following model: Sales = + Price + Advert
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.
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