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Linear Regression Model

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Added on  2020-02-05

Linear Regression Model

   Added on 2020-02-05

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a.GivenQt=αPBtβ2PLtβ3PRtβ4Itβ5exp{et}etN(0,σ2)The model may be rewritten in linear form (by applying logs) as: logQt=logα+β2logPBt+β3logPLt+β4logPRt+β5logIt+logetUsing the data provided, the log-log linear model is estimated using OLS as; SUMMARY OUTPUTRegression StatisticsMultiple R0.908508994R Square0.825388591Adjusted R Square0.797450766Standard Error0.026046069Observations30ANOVAdfSSMSFSignificance FRegression40.0801696920.02004229.543773.8E-09Residual250.0169599420.000678Total290.097129635CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%Intercept-1.408520231.625564037-0.866480.394467-4.756431.939392-4.756431.939392Log PB-1.020418770.23904185-4.268790.000248-1.51273-0.5281-1.51273-0.5281Log PL -0.582933650.560150059-1.040670.307987-1.736580.570717-1.736580.570717log PR0.2095449720.0796926342.6294150.0144220.0454150.3736750.0454150.373675Log I0.9228637520.4155141822.2210160.0356390.0670961.7786310.0670961.778631logQt=log1.40851.0204logPBt0.5829logPLt+0.2095logPRt+0.9229logItSE1.62550.23900.56020.079690.4155b.Interpreted the regression coefficients From the estimated model α = -1.4085, implying that the average quantity of beer consumed is negative, if all other factors include in the model remain at zero. But then again the value isnot statistically significant it has a p-value of 0.39 which is larger than the significance level (0.05)Holding other factors constant, a 100 % rise in the price of beer would reduce beer consumption by 102.04% and vice versa. The value is statistically different from zero. Consequently, ceteris paribus a 100% change in PR and income would change alcohol consumption by 20.95% and 92.28% respectively. These values are also statistically
Linear Regression Model_1
significant. Note that β3 has p-value larger than significant level and hence has its estimated value does not have statistical significance. However, the negative sign implies that a rise in price liquor ma lead to a reduction in beer consumption. c.Overall significanceFrom t the output table, the significance F = 0.0000000038 which is less than the significance level (0.05), hence at least of non-constant coefficients is not zero and hence the model is significant at 95% confidence level. d. Test the hypothesis H0=β3=β4=0 against HA=eitherβ3β4botharenotzero If the β3=β4=0 then the model would reduce from logQt=logα+β2logPBt+β3logPLt+β4logPRt+β5logIt+loget.................(UR)TologQt=logα+β2logPBt+β5logIt+loget...............................Restrictedmodel(R)Table 2(restricted model)SUMMARY OUTPUTRegression StatisticsMultiple R0.88017R Square0.774699Adjusted R Square0.75801Standard Error0.028469Observations30ANOVAdfSSMSFSignificance FRegression20.0752460.03762346.419881.83E-09Residual270.0218830.00081Total290.09713CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%Intercept-2.902421.527086-1.900620.068077-6.035740.230905-6.035740.230905Log I1.1603860.3624633.2013920.0034870.4166731.9040980.4166731.904098Log PB-1.221250.231322-5.279421.44E-05-1.69588-0.74661-1.69588-0.74661Using the F –test
Linear Regression Model_2
Fstatistic=(RUR2RR2)q(1RUR2)(nk)Where n = number of observation = 30k = is the degrees of freedom = 4q= number of restrictions = 2 R2UR is the R-Squared of the unrestricted model = 0.8254 (from table 1)R2R is the R-Squared of the restricted model = 0.7746 (from table 2)Fstatistic=0.82540.77462(10.8254304)=3.7824F critical F2,260.05=3.3690The F statistic is greater than the critical F, and hence the hypothesis is rejected, implying that both β3β4 cannot be statistically equal to zero. At least one of the two is not zero, and hence the unrestricted model is better than the restricted model. e.Test the hypothesis H0:β2+β3+β4=β5=0 against HA: at least one of the coefficients is not zero If the β2+β3+β4=β5=0 then the model would reduce to logQt=logαAnd q =k = 4The f value Fvalue=R2k(1R2)(nk)=0.82544(10.8254304)=30.72Note that the F calculated value may also be obtained from table 1 (F ≈ 29.54376).F critical F2,260.05=3.3690
Linear Regression Model_3

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