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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics in a Global Economy Chapter 4 DEMAND ESTIMATION

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Scatter Diagram Regression Analysis

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Regression Line: Line of Best Fit Regression Line: Minimizes the sum of the squared vertical deviations (e t ) of each point from the regression line. Ordinary Least Squares (OLS) Method

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University REGRESSION ANALYSIS Given the following demand function: Y = A + B1 X + B2 P + B3 I + B4 Pr; X = selling expenses (advertising) Pr = price of substitutes What we want are estimates of the values of A, B1, B2, B3, & B4. Regression analysis describes the way in which one variable is related to another. It derives an equation that can be used to estimate the unknown value of one variable on the basis of the known value of the other variable(s).

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Simple Regression Model. the simple regression model takes the following form: Y i = A + B X i + e i ; Regression analysis assumes that the mean value of Y, given the value of X, is a linear function of X. In other words, the mean value of the dependent variable is assumed to be a linear function of the independent variable. Yi is the ith observed value of the dependent variable and Xi is the ith observed value of the independent variable. Essentially e i is an error term, that is, a random amount that is added to A+BX i (or subtracted if e i is negative).

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Because of the presence of the error term, the observed values of Y i fall around the population regression line (A+BX i ), not on it Regression analysis assumes that the values of e i are independent and that their mean value equals zero.

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Sample Regression Line (based on a sample). sample regression line (estimated regression line) describes the average relationship between the dependent variable and independent variable. The general expression of the sample regression line is: i.e, the value of the dependent variable predicted by the regression line, a & b = estimators of A and B. a = the intercept of the regression line b = the slope of the line, measure the change in the predicted value of Y associated with a one unit increase in X.

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Method of Least Squares. Used to determine the values of a and b. Since the deviation of the ith observed value of Y from the regression line equals, the sum of the squared deviations equals: Where n is the sample size. Using minimization technique we can find the values of a and b that minimize this expression, by differentiating these expression with respect to a and b and by setting these partial derivatives equal to zero.

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University (1) (2) solving equations (1) and (2) simultaneously, and letting equal the mean value of X in the sample and equal the mean of Y, we find that; and and

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University given the following data for company X. Given the following results of the table below = 2.533 + 1.504X; = 2.533 + 1.504X; if Y = the observed value of sales = the computed (estimated) value of sales based on the regression line. = the computed (estimated) value of sales based on the regression line. from the table Y = 4 when X = 1. But using the regression line: = 2.533 + 1.504(1) = 4.037 (Note there is a difference between the observed sales (4) and the estimated sales (4.037). = 2.533 + 1.504(1) = 4.037 (Note there is a difference between the observed sales (4) and the estimated sales (4.037).

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University if X = 0 then Y = 2.533 +1.504(0) = 2.533 (the intercept: the value of Y that intersects the vertical axis) Interpretation: if the firm’s selling expenses = 0, sales would be 2.533 million of units, and estimated sales go up 1.504 million units when selling expenses increase by 1m.

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Ordinary Least Squares Estimation Using Excel The model: Objective: Determine the slope and intercept that minimize the sum of the squared errors.

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Estimation Procedure

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University

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Tests of Significance Standard Error of the Slope Estimate A measure of the amount of scatter of individual observations about the regression line. It is useful in constructing prediction intervals - that is, intervals within which there is a specified probability that the dependent variable will lie.

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University if probability is set at 0.95, a very approximate prediction interval is: 2se; since se = 0.3702 if the predicted value of Y is 11, there is a probability that the firm’s sales will be between: 10.26 (11 – (2 × 0.37)) and 11.74 (11 + (2 × 0.37)) Example Calculation

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University The t-statistic (significance of individual variables). Managers need to know whether a particular independent variable influences the dependent variable. The least square estimates of B’s by chance may be positive even if their true values are zero. e.g., B1 = 1.76 i.e., selling expenses have an effect on sales (t=0.0001). To test whether the true value of B1 is zero we must look at the t-statistic of B1. The t-statistic has a distribution called t-distribution. All things equal, the bigger the value of t-statistic (in absolute terms), the smaller the probability that the true value of the regression coefficient in question is zero. In our case, there is only 1 in 10 000 that chance alone would have resulted in a large t-statistic.

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Calculation of the t Statistic Degrees of Freedom = (n-k) = (10 - 2) = 8 Critical Value at 5% level =2.306 ( is significant )

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Decomposition of Sum of Squares Total Variation = Explained Variation + Unexplained Variation Coefficient of Determination Coefficient of Correlation

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Decomposition of Sum of Squares

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Multiple Regression Analysis Model: Adjusted Coefficient of Determination Analysis of Variance and F Statistic

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Problems in Regression Analysis Multicollinearity. A situation in which two or more independent variables are very highly correlated. Under perfect linear correlation it is impossible to estimate the regression coefficients. e.g., (perfect linear correlation) Y = A + B1 X1i + B2 X2i; where X1i = 3X2i-1 or X1i = 6 + X2ior X1i = 2 + 4X2i; imperfect linear correlation. Y = A + B1 X1i + B2 X2i; where; X1 = price, X2 = nominal income (p.Q)

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University If two independent variables move together in a rigid fashion, there is no way to tell how much effect each has separately, all what we can observe is the effect of both combined. Consequences of mutlicollinearity - High R2 with no significant t-scores - High simple correlation coefficients (cross correlation matrix) How to deal with multicollinearity - Drop one or more of the multicollinear variables - Transform the multicollinear variables (e.g. first difference) - Increase sample size

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Serial Correlation (or Autocorrelation) Error terms are not independent, if this year’s error term is positive, next year is always positive ( positive serial correlation ), and if this year’s error term is negative, next year’s is always negative. This is a violation of the assumptions underlying regression analysis. [ should be E(reiej) = 0, if not, the simple correlation between two observations of the error term is not equal to zero] Consequences of Serial Correlation - Increases of the variances of the distributions - Leads to underestimate the standard errors of the coefficients.

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Detecting Serial correlation. - Durbin Watson Test Compare the computed DW with the DW tables to show whether d is so high or so low, that the hypothesis that there is no serial correlation should be rejected. if d < dL reject the hypothesis of no serial correlation. if d > du accept the hypothesis of no serial correlation if dL d du, the test is inconclusive. e.g.; if the hypothesis is that there is a negative serial correlation, we should - reject the hypothesis of no serial correlation if d<4-dL - accept the hypothesis of no serial correlation if d<4-du - if 4-du d 4-dL the test is inconclusive.

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University How to deal with serial correlation - take the difference of the variables - use generalized least squares

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Steps in Demand Estimation 1. Model Specification: Identify Variables - Identify the independent variables (in reality an empirical issue) 2. Specify Functional Form - Specify the mathematical form of the equation relating the mean value of the dependent variable to those of the independent variables. e.g., Y = f(X,P). This can take the following forms: Y = A + B1 Xi + B2 Pi + ei; B1>0, B2 0, B2<0or: log Y = log A + B1 log Xi + B2 log Xi + log ei;

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University 3. Collect your data. Data can be: - time series - cross section - cross section/time series (panel) 4. Estimate The Function 5. Test the Results

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Managerial Economics Prof. M. El-Sakka CBA. Kuwait University Managerial Economics Prof. M. El-Sakka CBA. Kuwait University

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