Estimating Demand Outline Where do demand functions come from? Sources of information for demand estimation Cross-sectional versus time series data Estimating a demand specification using the ordinary least squares (OLS) method. Goodness of fit statistics.

The goal of forecasting To transform available data into equations that provide the best possible forecasts of economic variables—e.g., sales revenues and costs of production—that are crucial for management.

Demand for air travel Houston to Orlando Q = Y + P O – 2P Recall that our demand function was estimated as follows: [4.1] Where Q is the number of seats sold; Y is a regional income index; P 0 is the fare charged by a rival airline, and P is the airline’s own fare. Now we will explain how we estimated this demand equation

Questions managers should ask about a forecasting equations 1.What is the “best” equation that can be obtained (estimated) from the available data? 2.What does the equation not explain? 3.What can be said about the likelihood and magnitude of forecast errors? 4.What are the profit consequences of forecast errors?

How do get the data to estimate demand forecasting equations? Customer surveys and interviews. Controlled market studies. Uncontrolled market data.

Campbell’s soup estimates demand functions from data obtained from a survey of more than 100,000 consumers

Survey pitfalls Sample bias Response bias Response accuracy Cost

Time -series data: historical data--i.e., the data sample consists of a series of daily, monthly, quarterly, or annual data for variables such as prices, income, employment, output, car sales, stock market indices, exchange rates, and so on. Cross-sectional data: All observations in the sample are taken from the same point in time and represent different individual entities (such as households, houses, etc.) Types of data

Time series data: Daily observations, Korean Won per dollar

Student IDSexAgeHeightWeight M216’1”178 lbs M285’11”205 lbs F195’8”121 lbs F225’4”98 lbs M206’2”183 lbs Example of cross sectional data

Estimating demand equations using regression analysis Regression analysis is a statistical technique that allows us to quantify the relationship between a dependent variable and one or more independent or “explanatory” variables.

Y X 0 X and Y are not perfectly correlated. However, there is on average a positive relationship between Y and X X1X1 X2X2 Regression theory

11 Y1Y1 E(Y|X 1 ) Y X0X1X1 E(Y |X i ) =  0 +  1 X i  1 = Y 1 - E(Y|X 1 ) We assume that expected conditional values of Y associated with alternative values of X fall on a line.

Our model is specified as follows: Q = f (P) where Q is ticket sales and P is the fare Specifying a single variable model Q is the dependent variable—that is, we think that variations in Q can be explained by variations in P, the “explanatory” variable.

 0 and  1 are called parameters or population parameters.  We estimate these parameters using the data we have available Estimating the single variable model [1] [2] Since the data points are unlikely to fall exactly on a line, (1) must be modified to include a disturbance term (ε i )

Estimated Simple Linear Regression Equation n The estimated simple linear regression equation is the estimated value of y for a given x value. is the estimated value of y for a given x value. b 1 is the slope of the line. b 1 is the slope of the line. b 0 is the y intercept of the line. b 0 is the y intercept of the line. The graph is called the estimated regression line. The graph is called the estimated regression line.

Estimation Process Regression Model y =  0 +  1 x +  Regression Equation E ( y ) =  0 +  1 x Unknown Parameters  0,  1 Sample Data: x y x 1 y x n y n b 0 and b 1 provide estimates of  0 and  1 Estimated Regression Equation Sample Statistics b 0, b 1

Least Squares Method Least Squares Criterion where: y i = observed value of the dependent variable for the i th observation for the i th observation^ y i = estimated value of the dependent variable for the i th observation for the i th observation

Slope for the Estimated Regression Equation Least Squares Method

n y -Intercept for the Estimated Regression Equation Least Squares Method where: x i = value of independent variable for i th observation observation n = total number of observations _ y = mean value for dependent variable _ x = mean value for independent variable y i = value of dependent variable for i th observation observation

Line of best fit The line of best fit is the one that minimizes the squared sum of the vertical distances of the sample points from the line

1.Specification 2.Estimation 3.Evaluation 4.Forecasting The 4 steps of demand estimation using regression

Table 4-2 Ticket Prices and Ticket Sales along an Air Route

Simple linear regression begins by plotting Q-P values on a scatter diagram to determine if there exists an approximate linear relationship:

Scatter plot diagram with possible line of best fit Average One-way Fare $ Demand curve: Q = 330- P Number of Seats Sold per Flight

Note that we use X to denote the explanatory variable and Y is the dependent variable. So in our example Sales ( Q ) is the “ Y ” variable and Fares ( P ) is the “ X ” variable. Q = Y P = X

Computing the OLS estimators We estimated the equation using the statistical software package SPSS. It generated the following output:

Reading the SPSS Output From this table we see that our estimate of  0 is and our estimate of  1 is –1.63. Thus our forecasting equation is given by:

Step 3: Evaluation Now we will evaluate the forecasting equation using standard goodness of fit statistics, including: 1.The standard errors of the estimates. 2.The t-statistics of the estimates of the coefficients. 3.The standard error of the regression (s) 4.The coefficient of determination (R 2 )

We assume that the regression coefficients are normally distributed variables. The standard error (or standard deviation) of the estimates is a measure of the dispersion of the estimates around their mean value. As a general principle, the smaller the standard error, the better the estimates (in terms of yielding accurate forecasts of the dependent variable). Standard errors of the estimates

The following rule-of-thumb is useful: The standard error of the regression coefficient should be less than half of the size of the corresponding regression coefficient.

Note that: Let denote the standard error of our estimate of  1 Thus we have: Where: and k is the number of estimated coefficients Computing the standard error of  1

By reference to the SPSS output, we see that the standard error of our estimate of  1 is 0.367, whereas the (absolute value)our estimate of  1 is 1.63 Hence our estimate is about 4 ½ times the size of its standard error.

The SPSS output tells us that the t statistic for the the fare coefficient (P) is –4.453 The t test is a way of comparing the error suggested by the null hypothesis to the standard error of the estimate.

ß To test for the significance of our estimate of  1, we set the following null hypothesis, H 0, and the alternative hypothesis, H 1 ßH 0 :  1  0 ßH 1 :  1 < 0 ßThe t distribution is used to test for statistical significance of the estimate: The t test

 The coefficient of determination, R 2, is defined as the proportion of the total variation in the dependent variable (Y) "explained" by the regression of Y on the independent variable (X). The total variation in Y or the total sum of squares (TSS) is defined as:  The explained variation in the dependent variable(Y) is called the regression sum of squares (RSS) and is given by: Note: Coefficient of determination (R 2 )

What remains is the unexplained variation in the dependent variable or the error sum of squares (ESS) We can say the following: TSS = RSS + ESS, or Total variation = Explained variation + Unexplained variation R 2 is defined as:

We see from the SPSS model summary table that R 2 for this model is.586

 Note that: 0  R 2  1  If R 2 = 0, all the sample points lie on a horizontal line or in a circle  If R 2 = 1, the sample points all lie on the regression line  In our case, R 2  0.586, meaning that 58.6 percent of the variation in the dependent variable (consumption) is explained by the regression. Notes on R 2

This is not a particularly good fit based on R 2 since 41.4 percent of the variation in the dependent variable is unexplained.

 The standard error of the regression (s) is given by: Standard error of the regression

 The model summary tells us that s = 18.6  Regression is based on the assumption that the error term is normally distributed, so that 68.7% of the actual values of the dependent variable (seats sold) should be within one standard error (  $18.6 in our example) of their fitted value.  Also, 95.45% of the observed values of seats sold should be within 2 standard errors of their fitted values (  37.2).

Step 4: Forecasting Recall the equation obtained from the regression results is : Our first step is to perform an “in-sample” forecast.

At the most basic level, forecasting consists of inserting forecasted values of the explanatory variable P (fare) into the forecasting equation to obtain forecasted values of the dependent variable Q (passenger seats sold).

In-Sample Forecast of Airline Sales

Our ability to generate accurate forecasts of the dependent variable depends on two factors: Do we have good forecasts of the explanatory variable? Does our model exhibit structural stability, i.e., will the causal relationship between Q and P expressed in our forecasting equation hold up over time? After all, the estimated coefficients are average values for a specific time interval ( ). While the past may be a serviceable guide to the future in the case of purely physical phenomena, the same principle does not necessarily hold in the realm of social phenomena (to which economy belongs). Can we make a good forecast?

Single Variable Regression Using Excel We will estimate an equation and use it to predict home prices in two cities. Our data set is on the next slide

CityIncomeHome Price Akron, OH Atlanta, GA Birmingham, AL Bismark, ND Cleveland, OH Columbia, SC Denver, CO Detroit, MI Fort Lauderdale, FL Hartford, CT Lancaster, PA Madison, WI Naples, FL Nashville, TN Philadelphia, PA Savannah, GA Toledo, OH Washington, DC Income (Y) is average family income in 2003 Home Price (HP) is the average price of a new or existing home in 2003.

Model Specification

Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations18 Coefficients Standard Errort Stat Intercept Income ANOVA dfSS Regression Residual Total Excel Output

CityIncomePredicted HP Meridian, MS59,600 $ 138, Palo Alto, CA121,000 $ 281, Equation and prediction