The Multiple Regression Model Onon Khanoi Berhanu Anbesa

The Multiple Regression Model Onon Khanoi Berhanu Anbesa

Content 1. Introduction 2. Estimating a Multiple Regression Model 3. Forecasting from a Multiple Regression Model 4. Interval Estimation 5. Hypothesis Testing 6. Polynomial Equations 7. Interaction Variables 8. Goodness-of-fit 9. Summary 2/31

Introduction 1 • Multiple regression model – econometric model with more than one explanatory variable. • Our model is: SALES = ß 1+ß 2 PRICE + ß 3 ADVERT+e , where SALES – monthly sales revenue in a given city (1000 USD) PRICE – price in that city ADVERT – monthly advertising expenditure in that city (1000 USD) ß 2 - the change in monthly SALES (1000 USD) when price index PRICE is increased by 1 unit ($1) and advertising expenditure ADVERT is held constant ß 3 - the change in monthly SALES (1000 USD) when advertising expenditure ADVERT is increased by 1 unit ($1000) and price index PRICE is held constant. e – random error term 3/31

Opening Eviews work file 1 1. Open Eviews work file - Andy Data series Page name Number of observation 2. Create objects as a group: Sales, Price and Advert 3. Select the name table 5_1 4/31

Spreadsheet 1 1. Click View → Descriptive Stats → Common Sample 5/31

Saving 1. Click Freeze to save 1 2. Click Name 2. Write name: Andy_statistics Table object 6/31

Estimating a Multiple Regression Model 2 Two ways: through Quick menu, Object menu and equation Command 1. Using the Quick menu write: sales c price advert Click Quick → Estimation Equation Explanatory variables Constant term Dependent variable Our equation is: SALES = ß 1+ß 2 PRICE + ß 3 ADVERT+e 7/31

Regression output 2 Name it burger_eqn Coefficient: 1. The negative coefficient on PRICE suggests that demand is price elastic. With advertising held constant, an increase in price of $1 will lead to a fall in monthly revenue of $7, 0908; or reduction in price of $1 will lead to an increase in revenue of $7, 908. 2. The coefficient on advertising is positive. With price held constant, an increase in advertising expenditure of $1000 will lead to an increase in sales revenue of $1, 863. 3. C is not directly interpretable. It is not possible that if both proce and advertising expenditure were zero the sales revenue would be $118, 914. 8/31

Forecasting from a Multiple Regression Model In addition to providing information about how sales change when price or advertising change, the estimated equation can be used for prediction. We will now forecast hamburger sales revenue for PRICE=5. 5 and ADVERT=1. 2 b 1=C(1) b 2=C(2) b 3=C(3) Objective is to get Eviews to perform the calculation: SALES=b 1 + b 2 x 5. 5+b 3 x 1. 2 9/31 3

Forecasting from a Multiple Regression Model: A Simple Forecasting Procedure Eviews command is: scalar sales_f=c(1)+c(2)*5. 5+c(3)*1. 2 You will see: Double click sales_f in work file Forecast result: 10/31 3

Forecasting from a Multiple Regression Model: Using the Forecast option We need to extend the size of work file to accommodate the observations for which we want forecast. Click Proc → Structure/Resize Current Page Change range of data to 76 Click Yes Range and sample changed to 76 Double click 11/31 3

Forecasting from a Multiple Regression Model: Using the Forecast option Open PRICE and change 76 to 5. 5 Click to open for editing Change NA to 5. 5 Click to close for editing New value 3 Same steps for ADVERT change to 1. 2 Click to open for editing Change NA to 1. 2 Click to close for editing New value 12/31

Forecasting from a Multiple Regression Model: Using the Forecast option Open burger_eqn equation and click Forecast Open salesf and sef as a group Change: Forecast value Add name for standard error Forecast for observation 76 Click OK SALES= 77. 6555 and se(f)=4. 942 Can be used to compute a forecast Interval as SALES ±t(1 -α/2, 72) x se(f). 13/31 3

Interval Estimation: 4 The least squares covariance matrix The variances and covariances of the least squares estimators give us information about the reliability of the estimators. The smaller their variances, the higher the probability that they will produce estimates “near” the true parameter values. Open burger_eqn and click View → Covariance Matrix Var(b 1) Cov(b 1, b 2) Cov(b 1, b 3) Cov(b 1, b 2) Var(b 2) Cov(b 2, b 3) Cov(b 1, b 3) Cov(b 2, b 3) Var(b 3) 14/31

Interval Estimation: 4 Computing interval estimates Type the command one by one : scalar tc=@qtdist(0. 975, 72) scalar beta 2_low=c(2)-tc*@stderrs(2) scalar beta 2_up=c(2)+tc*@stderrs(2) scalar beta 3_low=c(3)-tc*@stderrs(3) scalar beta 3_up=c(3)+tc*@stderrs(3) To view upper and lower bounds of the interval estimates Value BETA 2_LOW -10. 09268 BETA 2_UP -5. 723032 Easy way to get a same result: Open burger_eqn Click View → Coefficient Diagnostics → Confidence Intervals → choose. 95 → OK 15/31

Interval Estimation: Interval estimates for linear combinations of coefficients • • Big Andy plans to increase advertising expenditure by $800 and drop the price by 40 cents. λ=E(SALES 1)-E(SALES 0)= -0. 4ß 2+0. 8ß 3 1. Open Burger_eqn → click Coefficient Diagnostics → Wald Test – Coefficient Restrictions… 2. Type -0. 4*c(2)+0. 8*c(3)=0 Estimate 16/31 4

Hypothesis Testing : Two-tail tests of significance 5 Ho =ß 2=0 (no price effect) H 1 =ß 2≠ 0 (there is a price effect ) Ho =ß 3=0 (no advertising effect) H 1 =ß 3≠ 0 (there is an advertising effect ) T-values and p-values for two-tail tests of significance 17/31

Hypothesis Testing : Two-tail tests of significance 5 P-value for ADVERT is 0. 0080. We can confirm the p-value giving the command: scalar pval_advert=2*@ctdist(-2. 726, 72) Open pval_advert In the case of advertising expenditure we reject Ho =ß 3=0 at a 5% significance level because p-value of 0. 0080 is less than 0. 05. If we want to make a decision about Ho = by comparing the calculated value t=2. 726 to a 5% critical value. Give command: scalar tc=@qtdist(0. 975, 72) The answer is tc=1. 993, a value that leads us to reject Ho =ß 3=0 because 2. 726 > 1. 993 18/31

Hypothesis Testing : One-tail tests of significance 5 Ho =ß 2≥ 0 H 1 =ß 2<0 Eview command: scalar tc 1=@qtdist(0. 05, 72) Thus, making the test decision by reference to the critical value, we reject Ho =ß 2≥ 0 in favor of H 1 =ß 2<0 because -7. 215 < - 1. 666 19/31

Hypothesis Testing : Testing nonzero values 5 One tail: For advertising to be effective, ß 2 must be greater than 1. Thus we test: Ho =ß 3≤ 1 H 1 =ß 3>1. Eview command: • scalar t 3=(c(3)-1)/@stderrs(3) • scalar pval 3=1 -@ctdist(t 3, 72) result is t=1. 263 resultis p=0. 105 Choosing α=0. 05 as our significance level, relevant critical value is t(o. 95, 72)=1. 666. Since 1. 263<1. 666, we don’t reject Ho. There is insufficient evidence in our sample conclude that advertising will be cost effective. 20/31

Hypothesis Testing : Testing nonzero values 5 We will now test two-tails test: Ho =ß 3=1 H 1 =ß 3≠ 1 Open burger_eqn → Coefficient Diagnostics → Wald Test – Coefficient Restrictions Type null hypothesis c(3)=1 Or you can use this command: burger_eqn. wald c(3) 21/31

Hypothesis Testing : Testing nonzero values T-value numerator 5 T-value denominator t=b 3 -1/se(b 3) 22/31

Hypothesis Testing : Testing linear combinations of coefficients 5 Big Andy’s marketing advisor claims that dropping the price by 20 cents will be more effective for increasing sales revenue than increasing advertising expenditure by $500. -0. 2ß 2>0. 5ß 3. Hypothesis: Ho =-0. 2ß 2 -o. 5ß 3≤ 0 H 1 =-0. 2ß 2 -o. 5ß 3>0 Open burger_eqn → Coefficient Diagnostics → Wald Test – Coefficient Restrictions Write -0. 2*c(2)-0. 5*c(3)=0 Two-tail p -value As 1. 263<1. 666, we do not reject Ho. At a 5% significance level, there is not enough evidence to support marketing advisor. You can also use command: burger_eqn. wald -0. 2*c(2)-0. 5*c(3)=0 23/31

Polynomial Equations 6 To take into account the fact that the marginal effect of advertising is most likely a diminishing function of advertising, the squared value of advertising is added to Andy’s SALES equation. SALES = ß 1+ß 2 PRICE + ß 3 ADVERT+ß 4 ADVERT 2+e Click Quick →Equation Estimation → write sales c price advert^2 Name it andy_quad 24/31

Polynomial Equations 6 Example: the marginal effect of advertising on SALES when ADVERT=2 ($2000 per week). Open ANDY_QUAD → View →Coefficient Diagnostics → Wald Test-Coefficient Restrictions Or use command → andy_quad. wald c(3)+2*c(4)*2=0 Marginal effect when ADVERT=2 If ADVERT=2, the marginal effect of an additional $1000 of advertising expenditure is $1, 079 25/31

Polynomial Equations 6 The Wald test also reports the t –statistics for the hypothesis that the estimated marginal effect is zero: We cannot reject the null hypothesis that the marginal effect is zero at the 5% level of significance. From the figure you can see that 2 is near to the maximum point where the slope of the fitted quadratic function is zero. 26/31

Polynomial Equations The optimal level of advertising 6 Andy’s objective is maximize profit, but not SALES. Thus, advertising should be increased to the point where ß 3+2ß 4 ADVERT 0=1 Enter command: andy_quad. wald (1 -c(3))/(2*c(4))=0 27/31

Interaction Variables 7 Open new work file – pizza 4 Our equation is PIZZA = ß 1+ß 2 AGE + ß 3 INCOME+e As age might moderate the income effect: PIZZA = ß 1+ß 2 AGE + ß 3 INCOME+ß 4(AGEx. INCOME)+e Command: equation pizza_eq. Is pizza c age income age*income 28/31

Interaction Variables 7 For individuals of ages 20 and 50, the marginal effect are given by pizza_eq. wald c(3)+c(4)*20=0 pizza_eq. wald c(3)+c(4)*50=0 The results are: The marginal effect of income for the 50 -year-old is much smaller than the marginal effect of income for the 20 -year-old. 29/31

Goodness-of-fit 8 Below the parameter estimates are various summary measures including the R 2=0. 448258 Omitted the intercept: 30/31

Summary • • 9 Multiple regression model – econometric model with more than one explanatory variable. We did regression SALES = ß 1+ß 2 PRICE + ß 3 ADVERT+e We forecasted hamburger sales revenue for PRICE=5. 5 and ADVERT=1. 2 with SALES=b 1 + b 2 x 5. 5+b 3 x 1. 2 Interval estimation: – Computed Confidence Intervals with. 95 significance level. – Checked how Sales will change when advertising expenditure increased by $800 and the price dropped by 40 cents. • Checked hypothesis: 1. Ho =ß 2=0 H 1 =ß 2≠ 0 Ho =ß 3=0 H 1 =ß 3≠ 0 3. Ho =ß 3≤ 1 H 1 =ß 3>1. • • 4. Ho =ß 3=1 H 1 =ß 3≠ 1 2. Ho =ß 2≥ 0 H 1 =ß 2<0 5. Ho =-0. 2ß 2 -o. 5ß 3≤ 0 H 1 =-0. 2ß 2 -o. 5ß 3>0 Polynominal equation with SALES = ß 1+ß 2 PRICE + ß 3 ADVERT+ß 4 ADVERT 2+e Interaction variables with PIZZA = ß 1+ß 2 AGE + ß 3 INCOME+ß 4(AGEx. INCOME)+e 31