Chapter 5Estimating Demand Functions

Identification Problem

The inability to distinguish between moves

along a demand curve and shifts in supply and/or

demand that reflect changes in behavior

S99

D99

S98

D98 S97

D97

Quantity

Price

D|

D

Estimated Demand Contrasted with Actual Demand

Alternative Methods of Estimating Demand

Consumer interviews.Market experiments.Regression analysis.

Regression Analysis

A statistical technique that describes the way in which one variable is related to another

Used to estimate demandAssumes that the mean value of the dependent

variable is a linear function of the independent variable

Yi = A + BXi + ei

The Mean Value of X Falls on the Population Regression Line

Quantity

Price

A + BX

Population regression line

X1 X2

.

.e1 = -1

Y1A + BX1

.

.e2 = 1.5

Y2

A + BX2

Ŷ = the value of the dependent variable

predicted by the regression line

a = the y-intercept of the regression line

b = the slope of the regression line

Xt = the value of the independent variable

at t

Ŷ = a + bxt

Method of Least Squares

(Yi - Ŷ i)2 =

i = 1

n

i = 1

n

(Yi - a - bxi)2

Coefficient of Determination

Commonly called the “R-squared”measures goodness of fit of the estimated

regression linevaries between 0 and 1

x

y

x

y

Multiple Regression

Includes two or more independent variables

Yi = A + b1Xi + b2Pi + ei

where: Yi = sales

xi = selling expense

Pi = price

Regression Analysis

Given data on relevant variables, the empirical method of Regression Analysis can be used to estimate demand functions. We will begin by considering a simplified version of a demand function showing quantity demanded as a function of only price. Such a demand function would clearly be inadequate to successfully estimate a demand function. We use this simplified example of a demand function merely as a means of illustrating the technique of regression analysis. Once the technique is fully developed, we will proceed to estimate more realistic demand functions.

Quantity Demanded and Price

Let Yt be quantity demanded (i.e. Sales) in period t.

Let Xt be the price charged in period t. Plotting a firm’s historical data for these two variables we may have the graph shown on the right.

As the graph reveals, there is apparently a negative relationship between Sales and Price, i.e. as Price increases, Sales also tends to decrease.

Fitting A Line To Data

The relationship between Yt and Xt can be represented (albeit imperfectly) by a line that “best fits” the scatter of points. The line shown is defined by its intercept, a, and its slope, b. Notice, however, that the actual points do not fall exactly on the line. This is due to the fact that Sales may be partly determined by price, but other factors such as advertising, which are not taken into account, also affect Sales.

Fitting A Line To Data (continued)

Regression analysis is basically focused on methods of estimating lines that best fit data, such as the one shown in the previous slide.

Each dot shown in the graph can be broken up in to two pieces: the amount of Sales, , predicted by the line, and the resulting error (or “residual”) in prediction, et. Or, mathematically,

Or, since:

We have:

tY

tetYtY

tbXatY

tetbXatY

Method of Ordinary Least-Squares

Our goal is thus to estimate a line by finding a value for a and b such that resulting line “best fits” the data. We do so by finding values of a and b such that the sum of squared errors is minimized. That is, we can write the sum of squared errors as:

where n is the number of observations in our sample of data. This sum of squared errors can be minimized using differential calculus. Leaving out the details, minimizing this equation with respect to a and b yields the following results:

and

n

1t

2n

1t

2t )tbXatY(e

n

1t

2

n

1t

)XtX(

)YtY)(XtX(b XbYa

OLS Example: by-hand

2978.5533854.4

55.12922n

1t2)XtX(

n

1t)YtY)(XtX(

b -

97.16898

]107.3*55.12922[00.7645XbYa

ticePr*55.297897.16898tSales

Y (Sales) X (Price) (X - Xbar) (Y - Ybar) (X - Xbar)2 (X - Xbar)*(Y - Ybar)3160 4.24 1.133 -4485.000 1.284 -5082.0665868 2.96 -0.147 -1777.000 0.022 260.9975872 3.69 0.583 -1773.000 0.340 -1033.8815911 3.77 0.663 -1734.000 0.440 -1149.8596134 2.82 -0.287 -1511.000 0.082 433.4686216 3.59 0.483 -1429.000 0.233 -690.3867476 2.89 -0.217 -169.000 0.047 36.6527950 3.64 0.533 305.000 0.284 162.6038038 2.6 -0.507 393.000 0.257 -199.2028253 3.23 0.123 608.000 0.015 74.8608429 3.07 -0.037 784.000 0.001 -28.9108862 2.77 -0.337 1217.000 0.113 -409.9779240 2.73 -0.377 1595.000 0.142 -601.1169348 2.54 -0.567 1703.000 0.321 -965.388

10079 2.91 -0.197 2434.000 0.039 -479.19411484 2.26 -0.847 3839.000 0.717 -3251.153

Sum: 122320.000 49.710 4.339 -12922.550Average: 7645.000 3.107

OLS Example: using Excel

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.784154618R Square 0.614898466Adjusted R Square 0.587391213Standard Error 1312.194257Observations 16

ANOVAdf SS MS F Significance F

Regression 1 38490403.26 38490403.26 22.35404887 0.000323603Residual 14 24105952.74 1721853.767Total 15 62596356

Coefficients Standard Error t Stat P-valueIntercept 16898.96858 1984.567282 8.515190554 6.57413E-07Price -2978.545509 629.9791004 -4.72800686 0.000323603

Interpreting The OLS Output

R Square: A measure of the “goodness of fit.” It is the proportion of the variation in the dependent variable explained by the regression. It has a maxmimum of 1 (100%) meaning an perfect fit, and a minimum of 0 (0%), meaning no fit at all. Thus, the closer to 1, the better the fit.

Standard Error: The size of the typical error (et).

Significance F This shows the probability of obtaining the estimated values of the X coefficients from a sample regression if it were true that all the population's X coefficients are simultaneously equal to zero. If the Significance F is less than our chosen level of significance, then we can conclude that the model as a whole is statistically significant in explaining the values of the dependent variable

Interpreting The OLS Output

Coefficients: The estimated value for a and b for our model. The “Intercept” is the estimate for a. The coefficient for “Price” is our estimate for b.

P-Value: Used in hypothesis testing to test whether a sample estimate of a coefficient is statistically different from zero. It represents the probability of achieving the estimated coefficient for the sample at hand, if in fact the population's coefficient were zero. If the p-value is less than the chosen level of significance, then the zero hypothesis is rejected.

Forecasting Using Regression Output

Given an estimated regression, we can use it to forecast or make predictions. For example, suppose we have our estimated equation:

If we decide to seta price of $4 next period, we can predict sales as:

ticePr*55.297897.16898tSales

77.4984)4(*55.297897.16898tSales

Forecasting Using Regression Output (Continued)

Interval estimates: The previous forecast was a “point estimate” (i.e. an estimated number). As an alternative, we can estimate an interval that we are reasonably sure will contain the actual value of sales given advertising is $425. To do so we take our point estimate and add and subtract 2 times the standard error, thus creating an approximate 95% confidence interval*:

Or, in this case,

Or, the 95% confidence interval is approximately: (2360.38, 7609.16).

[*It should be noted that this interval estimate is only reliable if the independent variable’s value (i.e. the value chosen for Price) is close to its mean value. Values not close to their mean require a more complicated formula.]

)errordardtans(*2tY

39.262477.4984)19.1312(*2tSales

Multiple Regression Analysis

We may consider models with multiple independent variables. That is, we can consider advertising along with price, income, price of related goods, etc. Our regression model becomes:

Salest = a + b1Pricet + b2Advertisingt + b3Incomet + b4PRGt + et

Given data on the 4 independent variables, we can estimate values for a and b1 through b4 using the method of ordinary least-squares. Doing so by-hand is difficult, but computer programs such as Excel can estimate such regressions quite easily.

Example: Multiple Regression

Period Sales (Y) Price Advertising Income PRG1 11484 2.26 170 158.11 3.492 9348 2.54 165 173.36 2.853 8429 3.07 155 165.26 4.064 10079 2.91 175 172.92 3.645 9240 2.73 170 178.46 3.216 8862 2.77 125 198.62 3.667 6216 3.59 142 186.28 3.768 8253 3.23 165 188.98 3.499 8038 2.6 145 180.49 3.13

10 7476 2.89 138 183.33 3.211 5911 3.77 100 181.87 3.6512 7950 3.64 150 185 3.613 6134 2.82 138 184 2.9414 5868 2.96 100 188.2 3.1215 3160 4.24 125 175.67 3.5816 5872 3.69 100 188 3.53

Y=quantity of roses soldAdvertising = dollars of weekly advertising.Price= avg. wholesale price of roses, $/dozIncome = avg. weekly family income, $/weekPRG = avg. wholesale price of carnations, $/doz

Example: Multiple Regression with Excel

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.928986435R Square 0.863015797Adjusted R Square 0.81320336Standard Error 882.9038823Observations 16

ANOVAdf SS MS F Significance F

Regression 4 54021644.08 13505411.02 17.32530756 0.000102588Residual 11 8574711.919 779519.2654Total 15 62596356

Coefficients Standard Error t Stat P-valueIntercept 2511.188963 6746.291087 0.372232525 0.716791469Price -2926.923824 586.3890972 -4.991436297 0.000408005Advertising 31.03452082 11.87280417 2.613916677 0.024088631Income 11.21173383 27.74614421 0.404082591 0.693897098PRG 2276.845483 841.8340953 2.704624932 0.020486731

Predicted Equation

Using the coefficients from our regression, we have the following predicted equation:

Given values for the independent variables, we can predict values for Sales. We can also consider the statistical importance of the independent variables as we did earlier.

)tG2276.84(PR)tme11.21(Inco

)trtising31.03(Adve)tice2926.92(Pr-2511.19tSales

Goodness of Fit in Multiple Regression Models

Notice that the R Square increased from about 0.615 for our model with only Price, to about 0.863 for our model that adds Advertising, Income and PRG. Note, however, that to compare two regressions with the same dependent variable (Sales) but differing numbers of independent variables, we must use the Adjusted R Square to determine which model is performing better. Thus, the model with just Price has an Adjusted R Square of 0.587, whereas our multiple regression model has an Adjusted R Square of 0.813. Thus our multiple regression model indeed seems to be outperforming our model with just Price as an independent variable.

Estimating Elasticities

Elasticities at the “means” Mean values:

Price= 3.11 Advertising=141.44 Income=180.53 PRG=3.4

predicted sales = 7630.65

Estimating Elasticities (continued)

Price:

Advertising:

Income:

PRG (cross):

19.165.7630

11.3*)92.2926(

QP

PQ

D

D

02.165.7630

43.3*)84.2276(Q

PPQ

D

RG

RG

Dcross

26.065.7630

53.180*)21.11(

QI

IQ

D

DI

57.065.7630

44.141*)03.31(

QA

AQ

D

DA

Estimating Elasticities (continued)

Constant Elasticities: Log Transformations:

Regressing the natural log of the dependent variable (sales) on the natural logs of the independent variables produces coefficients that are estimates of constant elasticities.

Estimating Elasticities (continued)

Period Ln Sales Ln Price Ln Advertising Ln Income Ln PRG1 9.349 0.815 5.136 5.063 1.2502 9.143 0.932 5.106 5.155 1.0473 9.039 1.122 5.043 5.108 1.4014 9.218 1.068 5.165 5.153 1.2925 9.131 1.004 5.136 5.184 1.1666 9.090 1.019 4.828 5.291 1.2977 8.735 1.278 4.956 5.227 1.3248 9.018 1.172 5.106 5.242 1.2509 8.992 0.956 4.977 5.196 1.141

10 8.919 1.061 4.927 5.211 1.16311 8.685 1.327 4.605 5.203 1.29512 8.981 1.292 5.011 5.220 1.28113 8.722 1.037 4.927 5.215 1.07814 8.677 1.085 4.605 5.238 1.13815 8.058 1.445 4.828 5.169 1.27516 8.678 1.306 4.605 5.236 1.261

Estimating Elasticities (continued)

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.898004337R Square 0.806411788Adjusted R Square 0.736016075Standard Error 0.157671294Observations 16

ANOVAdf SS MS F Significance F

Regression 4 1.139136866 0.284784217 11.45541043 0.000650212Residual 11 0.273462605 0.024860237Total 15 1.412599472

Coefficients Standard Error t Stat P-valueIntercept 0.288472593 5.315801042 0.054267003 0.9576955Ln Price -1.573628926 0.339585147 -4.633974535 0.000723708Ln Advertising 0.537498781 0.271145282 1.982327622 0.07297832Ln Income 1.179598527 0.88303153 1.335850971 0.208578215Ln PRG 1.29888021 0.519122476 2.502068913 0.029398065

Standard error of estimate

A measure of the amount of scatter of individual observations about the regression

lineUseful for constructing prediction intervals

Y-hat +/- 2 standard errors

F-statistic

Tells us whether or not the group of independent variables explains a statistically significant portion of the variation in the dependent variable.

Large values of the F-statistic indicate that at least one of the independent variables is helping to explain variation in the dependent variable.

t-statistic

Tells us whether or not each particular independent variable explains a statistically significant portion of the variation in the dependent variable

All else equal, larger values for the t-stat are betterRule of Thumb: as long as the t-stat is greater than

2, the independent variable belongs in the equation

Multicollinearity

A situation in which two or more of the independent variables are highly correlated

Indicated by high r-squared and significant F-stat, but low t-stats for the independent variables

Serial Correlation

A situation in which error terms are not independent of one another over time

To detect, consider the Durbin-Watson statistic

Rule of Thumb: If the DW is close to 2, serial correlation is not present. If the DW is close to 1 or 4, serial correlation is present

Further Analysis of Residuals

Plot scatters of the residuals against each independent variable

There are problems with the regression analysis if patterns emerge