Applied Business Forecasting and planning Multiple Regression Analysis Introduction In simple linear regression we studied the relationship between one explanatory variable and one response variable. Now, we look at situations where several explanatory variables works together to explain the response. Introduction Following our principles of data analysis, we look first at each variable separately, then at relationships among the variables. We look at the distribution of each variable to be used in multiple regression to determine if there are any unusual patterns that may be important in building our regression analysis. Multiple Regression Example. In a study of direct operating cost, Y, for 67 branch offices of consumer finance charge, four independent variables were considered:
X1: Average size of loan outstanding during the year, X2 : Average number of loans outstanding, X3 : Total number of new loan applications processed, and X4 : Office salary scale index. The model for this example is Y 0 1 x1 2 x2 3 X 3 4 x4 Formal Statement of the Model General regression model Y 0 1 x1 2 x2 k xk 0, 1, , k are parameters X1, X2, ,Xk are known constants
, the error terms are independent N(o, 2) Estimating the parameters of the model The values of the regression parameters i are not known. We estimate them from data. As in the simple linear regression case, we use the least-squares method to fit a linear function y b0 b1 x1 b2 x2 bk xk to the data. The least-squares method chooses the bs that make the sum of squares of the residuals as small as possible. Estimating the parameters of the model The least-squares estimates are the values that minimize the quantity n 2 ( y y )
i i i 1 Since the formulas for the least-squares estimates are complicated and hand calculation is out of question, we are content to understand the least-squares principle and let software do the computations. Estimating the parameters of the model The estimate of i is bi and it indicates the change in the mean response per unit increase in Xi when the rest of the independent variables in the model are held constant. The parameters i are frequently called partial regression coefficients because they reflect the partial effect of one independent variable when the rest of independent variables are included in the model and are held constant Estimating the parameters of the model The observed variability of the responses about this fitted model is measured by the variance n 1 S ( yi y i ) 2
n k 1 i 1 2 and the regression standard error s s2 Estimating the parameters of the model In the model 2 and measure the variability of the responses about the population regression equation. It is natural to estimate 2 by s2 and by s. Analysis of Variance Table The basic idea of the regression ANOVA table are the same in simple and multiple regression. The sum of squares decomposition and the associated degrees of freedom are: 2 2 2 ( y y )
( y y ) ( y y ) i i i i SST df: SSR n 1 k (n k 1) SSE Analysis of Variance Table Source Sum of Squares
df Mean Square F-test Regression SSR k MSR= SSR/k MSR/MSE Error SSE n-k-1 MSE= SSE/n-k-1 Total SST n-1 F-test for the overall fit of the model
To test the statistical significance of the regression relation between the response variable y and the set of variables x1,, xk, i.e. to choose between the alternatives: H 0 : 1 2 k 0 H a : not all i (i 1, k ) equal zero We use the test statistic: MSR F MSE F-test for the overall fit of the model The decision rule at significance level is: Reject H0 if F F ( ; k , n k 1) Where the critical value F(, k, n-k-1) can be found from an F-table. The existence of a regression relation by itself does not assure that useful prediction can be made by using it.
Note that when k=1, this test reduces to the F-test for testing in simple linear regression whether or not 1= 0 Interval estimation of i For our regression model, we have: bi i s(bi ) has a t - distribution with n - k - 1 degrees of freedom Therefore, an interval estimate for i with 1- confidence coefficient is: Where bi t ( ; n k 1) s (bi ) 2 s (bi ) MSE ( x x )2 Significance tests for i To test: H 0 : i 0 H a : i 0
We may use the test statistic: Reject H0 if bi t s (bi ) t t ( ; n k 1) or 2 t t ( ; n k 1) 2 Multiple regression model Building Often we have many explanatory variables, and our goal is to use these to explain the variation in the response variable. A model using just a few of the variables often predicts about as well as the model using all the explanatory variables. Multiple regression model Building We may find that the reciprocal of a variable is a better choice than the variable itself, or that including the square of an explanatory variable
improves prediction. We may find that the effect of one explanatory variable may depends upon the value of another explanatory variable. We account for this situation by including interaction terms. Multiple regression model Building The simplest way to construct an interaction term is to multiply the two explanatory variables together. How can we find a good model? Selecting the best Regression equation. After a lengthy list of potentially useful independent variables has been compiled, some of the independent variables can be screened out. An independent variable May not be fundamental to the problem May be subject to large measurement error May effectively duplicate another independent variable in the list. Selecting the best Regression Equation. Once the investigator has tentatively
decided upon the functional forms of the regression relations (linear, quadratic, etc.), the next step is to obtain a subset of the explanatory variables (x) that best explain the variability in the response variable y. Selecting the best Regression Equation. An automatic search procedure that develops sequentially the subset of explanatory variables to be included in the regression model is called stepwise procedure. It was developed to economize on computational efforts. It will end with the identification of a single regression model as best. Example: Sales Forecasting Sales Forecasting Multiple regression is a popular technique for predicting product sales with the help of other variables that are likely to have a bearing on sales. Example The growth of cable television has created vast new potential in the home entertainment business. The following table gives
the values of several variables measured in a random sample of 20 local television stations which offer their programming to cable subscribers. A TV industry analyst wants to build a statistical model for predicting the number of subscribers that a cable station can expect. Example:Sales Forecasting Y = Number of cable subscribers (SUSCRIB) X1 = Advertising rate which the station charges local advertisers for one minute of prim time space (ADRATE) X2 = Kilowatt power of the stations non-cable signal (KILOWATT) X3 = Number of families living in the stations area of dominant influence (ADI), a geographical division of radio and TV audiences (APIPOP) X4 = Number of competing stations in the ADI (COMPETE) Example:Sales Forecasting The sample data are fitted by a multiple regression
model using Excel program. The marginal t-test provides a way of choosing the variables for inclusion in the equation. The fitted Model is SUBSCRIBE 0 1 ADRATE 2 APIPOP 3 COMPETE 4 SIGNAL Example:Sales Forecasting Excel Summary output SUMMARY OUTPUT Regression Statistics Multiple R 0.884267744 R Square 0.781929444 Adjusted R Square 0.723777295 Standard Error 142.9354188 Observations 20 ANOVA df Regression Residual Total Intercept AD_Rate Signal APIPOP Compete
4 15 19 SS 1098857.84 306458.0092 1405315.85 MS F Significance F 274714.4601 13.44626923 7.52E-05 20430.53395 Coefficients Standard Error t Stat P-value Lower 95%Upper 95% 51.42007002 98.97458277 0.51952803 0.610973806 -159.539 262.3795 -0.267196347 0.081055107 -3.296477624 0.004894126 -0.43996 -0.09443 -0.020105139 0.045184758 -0.444954014 0.662706578 -0.11641 0.076204 0.440333955 0.135200486 3.256896248 0.005307766 0.152161 0.728507 16.230071 26.47854322 0.61295181 0.549089662 -40.2076 72.66778 Example:Sales Forecasting
Do we need all the four variables in the model? Based on the partial t-test, the variables signal and compete are the least significant variables in our model. Lets drop the least significant variables one at a time. Example:Sales Forecasting Excel Summary Output SUMMARY OUTPUT Regression Statistics Multiple R 0.882638739 R Square 0.779051144 Adjusted R Square 0.737623233 Standard Error 139.3069743 Observations 20 ANOVA df Regression Residual Total Intercept AD_Rate
APIPOP Compete 3 16 19 SS 1094812.92 310502.9296 1405315.85 Coefficients Standard Error 51.31610447 96.4618242 -0.259538026 0.077195983 0.433505145 0.130916687 13.92154404 25.30614013 MS F Significance F 364937.64 18.80498277 1.69966E-05 19406.4331 t Stat 0.531983558 -3.36206646 3.311305499 0.550125146 P-value
0.602046756 0.003965102 0.004412929 0.589831583 Lower 95% Upper 95% -153.1737817 255.806 -0.423186162 -0.09589 0.15597423 0.711036 -39.72506442 67.56815 Example:Sales Forecasting The variable Compete is the next variable to get rid of. Example:Sales Forecasting Excel Summary Output SUMMARY OUTPUT Regression Statistics Multiple R 0.8802681 R Square 0.774871928 Adjusted R Square 0.748386273 Standard Error 136.4197776 Observations 20 ANOVA df
Regression Residual Total Intercept AD_Rate APIPOP 2 17 19 SS 1088939.802 316376.0474 1405315.85 MS 544469.901 18610.35573 F Significance F 29.2562866 3.13078E-06 Coefficients Standard Error t Stat P-value 96.28121395 50.16415506 1.919322948 0.07188916 -0.254280696 0.075014548 -3.389751739 0.003484198 0.495481252 0.065306012 7.587069489 7.45293E-07
Lower 95% Upper 95% -9.556049653 202.1184776 -0.41254778 -0.096013612 0.357697418 0.633265086 Example:Sales Forecasting All the variables in the model are statistically significant, therefore our final model is: Final Model SUBSCRIBE 96.28 0.25 ADRATE 0.495 APIPOP Interpreting the Final Model What is the interpretation of the estimated parameters. Is the association positive or negative? Does this make sense intuitively, based on what the data represents? What other variables could be confounders? Are there other analysis that you might consider doing? New questions raised? Multicollinearity
In multiple regression analysis, one is often concerned with the nature and significance of the relations between the explanatory variables and the response variable. Questions that are frequently asked are: What is the relative importance of the effects of the different independent variables? What is the magnitude of the effect of a given independent variable on the dependent variable? Multicollinearity Can any independent variable be dropped from the model because it has little or no effect on the dependent variable? Should any independent variables not yet included in the model be considered for possible inclusion? Simple answers can be given to these questions if The independent variables in the model are uncorrelated among themselves.
They are uncorrelated with any other independent variables that are related to the dependent variable but omitted from the model. Multicollinearity When the independent variables are correlated among themselves, multicollinearity or colinearity among them is said to exist. In many non-experimental situations in business, economics, and the social and biological sciences, the independent variables tend to be correlated among themselves. For example, in a regression of family food expenditures on the variables: family income, family savings, and the age of head of household, the explanatory variables will be correlated among themselves. Multicollinearity Further, the explanatory variables will also be correlated with other socioeconomic variables not included in the model that do affect family food expenditures, such as family size. Multicollinearity Some key problems that typically arise when the explanatory variables being considered for the regression model are highly correlated among themselves are:
1. 2. 3. Adding or deleting an explanatory variable changes the regression coefficients. The estimated standard deviations of the regression coefficients become large when the explanatory variables in the regression model are highly correlated with each other. The estimated regression coefficients individually may not be statistically significant even though a definite statistical relation exists between the response variable and the set of explanatory variables. Multicollinearity Diagnostics A formal method of detecting the presence of multicollinearity that is widely used is by the means of Variance Inflation Factor. It measures how much the variances of the estimated regression coefficients are inflated as compared to when the independent variables are not linearly related. 1 VIFj , 2 1 Rj
j 1,2, k R 2j Is the coefficient of determination from the regression of the jth independent variable on the remaining k-1 independent variables. Multicollinearity Diagnostics AVIF near 1 suggests that multicollinearity is not a problem for the independent variables. Its estimated coefficient and associated t value will not change much as the other independent variables are added or deleted from the regression equation. A VIF much greater than 1 indicates the presence of multicollinearity. A maximum VIF value in excess of 10 is often taken as an indication that the multicollinearity may be unduly influencing the least square estimates. the estimated coefficient attached to the variable is unstable and its associated t statistic may change considerably as the other independent variables are added or deleted. Multicollinearity Diagnostics
The simple correlation coefficient between all pairs of explanatory variables (i.e., X1, X2, , Xk ) is helpful in selecting appropriate explanatory variables for a regression model and is also critical for examining multicollinearity. While it is true that a correlation very close to +1 or 1 does suggest multicollinearity, it is not true (unless there are only two explanatory variables) to infer multicollinearity does not exist when there are no high correlations between any pair of explanatory variables. Example:Sales Forecasting Pearson Correlation Coefficients, N = 20 Prob > |r| under H0: Rho=0 SUBSCRIB ADRATE KILOWATT APIPOP COMPETE 1.00000 -0.02848 0.9051 0.44762 0.0478 0.90447 <.0001 0.79832
<.0001 -0.02848 0.9051 1.00000 -0.01021 0.9659 0.32512 0.1619 0.34147 0.1406 KILOWATT KILOWATT 0.44762 0.0478 -0.01021 0.9659 1.00000 0.45303 0.0449 0.46895 0.0370 APIPOP APIPOP
0.90447 <.0001 0.32512 0.1619 0.45303 0.0449 1.00000 0.87592 <.0001 COMPETE COMPETE 0.79832 <.0001 0.34147 0.1406 0.46895 0.0370 0.87592 1.00000 SUBSCRIB SUBSCRIB ADRATE ADRATE <.0001
Example:Sales Forecasting SUBSCRIBE 51.42 0.27 ADRATE - .02 SIGNAL 0.44 APIPOP 16.23COMPETE SUBSCRIBE 51.32 0.26 ADRATE 0.43 APIPOP 13.92COMPETE SUBSCRIBE 96.28 0.25 ADRATE 0.495 APIPOP Example:Sales Forecasting VIF calculation: Fit the model APIPOP 0 1 SIGNAL 2 ADRATE 3 COMPETE SUMMARY OUTPUT Regression Statistics Multiple R 0.878054 R Square 0.770978 Adjusted R Square 0.728036 Standard Error 264.3027 Observations 20 ANOVA df Regression Residual Total Intercept
Compete ADRATE Signal 3 16 19 SS MS 3762601 1254200 1117695 69855.92 4880295 Coefficients Standard Error t Stat -472.685 139.7492 -3.38238 159.8413 28.29157 5.649786 0.048173 0.149395 0.322455 0.037937 0.083011 0.457012 F Significance F 17.9541 2.25472E-05 P-value 0.003799 3.62E-05 0.751283 0.653806 Lower 95% Upper 95% -768.9402258 -176.43 99.86587622 219.8168
-0.268529713 0.364876 -0.138038952 0.213913 Example:Sales Forecasting Fit the model Compete 0 1 ADRATE 2 APIPOP 3 SIGNAL SUMMARY OUTPUT Regression Statistics Multiple R 0.882936 R Square 0.779575 Adjusted R Square 0.738246 Standard Error 1.34954 Observations 20 ANOVA df Regression Residual Total Intercept ADRATE Signal APIPOP SS MS F Significance F
3 103.0599 34.35329 18.86239 1.66815E-05 16 29.14013 1.821258 19 132.2 Coefficients Standard Error t Stat 3.10416 0.520589 5.96278 0.000491 0.000755 0.649331 0.000334 0.000418 0.799258 0.004167 0.000738 5.649786 P-value 1.99E-05 0.525337 0.435846 3.62E-05 Lower 95% Upper 95% 2.000559786 4.20776 -0.001110874 0.002092 -0.000552489 0.001221 0.002603667 0.005731 Example:Sales Forecasting Fit the model Signal 0 1 ADRATE 2 APIPOP 3 COMPETE SUMMARY OUTPUT Regression Statistics Multiple R
0.512244 R Square 0.262394 Adjusted R Square 0.124092 Standard Error 790.8387 Observations 20 ANOVA df Regression Residual Total Intercept APIPOP Compete ADRATE SS MS F Significance F 3 3559789 1186596 1.897261 0.170774675 16 10006813 625425.8 19 13566602 CoefficienSta ts ndard Error t Stat 5.171093 547.6089 0.009443 0.339655 0.743207 0.457012 114.8227 143.6617 0.799258 -0.38091 0.438238 -0.86919
P-value 0.992582 0.653806 0.435846 0.397593 Lower 95% Upper 95% -1155.707711 1166.05 -1.235874129 1.915184 -189.7263711 419.3718 -1.309935875 0.548109 Example:Sales Forecasting Fit the model ADRATE 0 1 Signal 2 APIPOP 3 COMPETE SUMMARY OUTPUT Regression Statistics Multiple R 0.399084 R Square 0.159268 Adjusted R Square 0.001631 Standard Error 440.8588 Observations 20 ANOVA df Regression Residual Total
Intercept Signal APIPOP Compete SS MS F Significance F 3 589101.7 196367.2 1.010346 0.413876018 16 3109703 194356.5 19 3698805 Coefficients Standard Error t Stat 253.7304 298.6063 0.849716 -0.11837 0.136186 -0.86919 0.134029 0.415653 0.322455 52.3446 80.61309 0.649331 P-value 0.408018 0.397593 0.751283 0.525337 Lower 95% Upper 95% -379.2865355 886.7474 -0.407073832 0.170329 -0.747116077 1.015175 -118.5474784 223.2367 Example:Sales Forecasting
VIF calculation Results: Variable ADRATE COMPETE SIGNAL APIPOP R- Squared 0.159268 0.779575 0.262394 0.770978 VIF 1.19 4.54 1.36 4.36 There is no significant multicollinearity. Qualitative Independent Variables Many variables of interest in business, economics, and social and biological sciences are not quantitative but are qualitative. Examples of qualitative variables are gender (male, female), purchase status (purchase, no
purchase), and type of firms. Qualitative variables can also be used in multiple regression. Qualitative Independent Variables An economist wished to relate the speed with which a particular insurance innovation is adopted (y) to the size of the insurance firm (x1) and the type of firm. The dependent variable is measured by the number of months elapsed between the time the first firm adopted the innovation and and the time the given firm adopted the innovation. The first independent variable, size of the firm, is quantitative, and measured by the amount of total assets of the firm. The second independent variable, type of firm, is qualitative and is composed of two classes-Stock companies and mutual companies. Indicator variables Indicator, or dummy variables are used to determine the relationship between qualitative independent variables and a dependent variable. Indicator variables take on the values 0 and 1. For the insurance innovation example, where the qualitative variable has two classes, we might define the indicator variable x2 as follows: 1 if stock company x2 0 otherwise Indicator variables
A qualitative variable with c classes will be represented by c-1 indicator variables. A regression function with an indicator variable with two levels (c = 2) will yield two estimated lines. Interpretation of Regression Coefficients In our insurance innovation example, the regression model is: y 0 1 x1 2 x2 Where: x1 size of firm 1 if stock company x2 0 otherwise Interpretation of Regression Coefficients To understand the meaning of the regression coefficients in this model, consider first the case of mutual firm. For such a firm, x2 = 0 and we have: y i b0 b1 x1 b2 (0) b0 b1 x1
Mutual firms For a stock firm x2 = 1 and the response function is: y i b0 b1 x1 b2 (1) (b0 b2 ) b1 x1 Stock firms Interpretation of Regression Coefficients The response function for the mutual firms is a straight line, with y intercept 0 and slope 1. For stock firms, this also is a straight line, with the same slope 1 but with y intercept 0+2. With reference to the insurance innovation example, the mean time elapsed before the innovation is adopted is linear function of size of firm (x1), with the same slope 1for both types of firms. Interpretation of Regression Coefficients 2 indicates how much lower or higher the
response function for stock firm is than the one for the mutual firm. 2 measures the differential effect of type of firms. In general, 2 shows how much higher (lower) the mean response line is for the class coded 1 than the line for the class coded 0, for any level of x1. Example: Insurance Innovation Adoption Here is the data set for the insurance innovation example: Months Elapsed 17 26 21 30 22 0 12 19 4 16 28 15 11 38 31 21 20 13 30 14 Size 151 92
175 31 104 277 210 120 290 238 164 272 295 68 85 224 166 305 124 246 type of firm 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1
1 1 1 1 Type Mutual Mutual Mutual Mutual Mutual Mutual Mutual Mutual Mutual Stock Stock Stock Stock Stock Stock Stock Stock Stock Stock Stock Example: Insurance Innovation Adoption Fitting the regression model y 0 1 x1 2 x2 Where
x1 size of firm 1 if stock company x2 0 otherwise fitted response function is: y 33.87 .1061x1 8.77 x2 Example: Insurance Innovation Adoption SUMMARY OUTPUT Regression Statistics Multiple R 0.95993655 R Square 0.92147818 Adjusted R Square 0.91224031 Standard Error 2.78630562 Observations 20 ANOVA df Regression Residual Total Intercept Size type of firm 2
17 19 SS MS F Significance F 1548.820517 774.4103 99.75016 4.04966E-10 131.979483 7.763499 1680.8 Coefficients Standard Error t Stat P-value 33.8698658 1.562588138 21.67549 8E-14 -0.10608882 0.007799653 -13.6017 1.45E-10 8.76797549 1.286421264 6.815789 3.01E-06 Lower 95% Upper 95% 30.57308841 37.16664321 -0.122544675 -0.089632969 6.053860079 11.4820909 Example: Insurance Innovation Adoption The fitted response function is: y 33.87 .1061x1 8.77 x2 Stock firms response function is: y (33.87 8.77) .1061x1
Mutual firms response function is: y 33.87 .1061x1 Interpretation ? Accounting for Seasonality in a Multiple regression Model Seasonal Patterns are not easily accounted for by the typical causal variables that we use in regression analysis. An indicator variable can be used effectively to account for seasonality in our time series data. The number of seasonal indicator variables to use depends on the data. If we have p periods in our data series, we can not use more than P-1 seasonal indicator variables. Example: Private Housing Starts (PHS) Housing starts in the United States measured in thousands of units. These data are plotted for 1990 Q1 through 1999Q4. There are typically few housing starts during the first quarter of the year
(January, February, March); there is usually a big increase in the second quarter of (April, May, June), followed by some decline in the third quarter (July, August, September), and further decline in the fourth quarter (October, November, December). Nov-98 Jul-98 1 Mar-98 Nov-97 Jul-97 1 Mar-97 Nov-96 Jul-96 Mar-96 Nov-95 Jul-95 1 Mar-95
Nov-94 100 Jul-94 250 Mar-94 Nov-93 Jul-93 1 Mar-93 Nov-92 Jul-92 200 Mar-92 Nov-91 Jul-91 Mar-91 Nov-90 Jul-90
Mar-90 Example: Private Housing Starts (PHS) Private Housing Starts (PHS) in Thousands of Units 400 350 300 1 1 1 150 1 "1" marks the first quarter of each year. 50 0 Example: Private Housing Starts (PHS) To Account for and measure this seasonality in a regression model, we will use three dummy variables: Q2 for the second quarter, Q3 for the third quarter, and Q4 for the fourth quarter. These will be coded as follows:
Q2 = 1 for all second quarters and zero otherwise. Q3 = 1 for all third quarters and zero otherwise Q4 = 1 for all fourth quarters and zero otherwise. Example: Private Housing Starts (PHS) Data for private housing starts (PHS), the mortgage rate (MR), and these seasonal indicator variables are shown in the following slide. Examine the data carefully to verify your understanding of the coding for Q2, Q3, Q4. Since we have assigned dummy variables for the second, third, and fourth quarters, the first quarter is the base quarter for our regression model. Note that any quarter could be used as the base, with indicator variables to adjust for differences in other quarters. Example: Private Housing Starts (PHS) PERIOD 31-M ar-90 30-Jun-90 30-Sep-90 31-Dec-90 31-M ar-91
30-Jun-91 30-Sep-91 31-Dec-91 31-M ar-92 30-Jun-92 30-Sep-92 31-Dec-92 31-M ar-93 30-Jun-93 30-Sep-93 31-Dec-93 31-M ar-94 30-Jun-94 30-Sep-94 31-Dec-94 31-M ar-95 30-Jun-95 30-Sep-95 31-Dec-95 31-M ar-96 30-Jun-96 30-Sep-96 31-Dec-96 31-M ar-97 30-Jun-97 30-Sep-97 31-Dec-97 31-M ar-98 30-Jun-98
30-Sep-98 31-Dec-98 31-M ar-99 30-Jun-99 30-Sep-99 31-Dec-99 PHS 217 271.3 233 173.6 146.7 254.1 239.8 199.8 218.5 296.4 276.4 238.8 213.2 323.7 309.3 279.4 252.6 354.2 325.7 265.9 214.2 296.7 308.2 257.2 240 344.5 324
252.4 237.8 324.5 314.6 256.8 258.4 360.4 348 304.6 294.1 377.1 355.6 308.1 MR 10.1202 10.3372 10.1033 9.9547 9.5008 9.5265 9.2755 8.6882 8.7098 8.6782 8.0085 8.2052 7.7332 7.4515 7.0778 7.0537 7.2958 8.4370 8.5882 9.0977 8.8123
7.9470 7.7012 7.3508 7.2430 8.1050 8.1590 7.7102 7.7905 7.9255 7.4692 7.1980 7.0547 7.0938 6.8657 6.7633 6.8805 7.2037 7.7990 7.8338 Q2 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0
0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 Q3 0 0 1 0 0 0 1 0 0
0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 Q4 0 0 0
1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
1 Example: Private Housing Starts (PHS) The regression model for private housing starts (PHS) is: PHS 0 1 ( MR) 2 (Q 2) 3 (Q3) 4 (Q 4) In this model we expect b1 to have a negative sign, and we would expect b2, b3, b4 all to have positive signs. Why? Regression results for this model are shown in the next slide. Example: Private Housing Starts (PHS) SUMMARY OUTPUT Regression Statistics Multiple R 0.885398221 R Square 0.78393001 Adjusted R Square 0.759236296 Standard Error 26.4498851 Observations 40 ANOVA df Regression Residual
Total Intercept MR Q2 Q3 Q4 4 35 39 SS MS F Significance F 88837.93624 22209.48406 31.74613731 3.33637E-11 24485.87476 699.5964217 113323.811 Coefficients Standard Error 473.0650749 35.54169837 -30.04838192 4.257226391 95.74106935 11.84748487 73.92904763 11.82881519 20.54778131 11.84139803 t Stat 13.31014264 -7.058206249
8.081130334 6.249911462 1.73524961 P-value 2.93931E-15 3.21421E-08 1.6292E-09 3.62313E-07 0.091495355 Lower 95% 400.9115031 -38.69102153 71.689367 49.91524679 -3.491564078 Upper 95% 545.2186467 -21.40574231 119.7927717 97.94284847 44.5871267 Example: Private Housing Starts (PHS) Use the prediction equation to make a forecast for each of the fourth quarter of 1999. Prediction equation: PHS 473.06 30.05( MR ) 95.74(Q 2) 73.93(Q3) 20.55(Q 4)
Example: Private Housing Starts (PHS) 400 350 Private Housing Starts (PHS) with a Simple Regression Forecast (PHSF1) and a Multiple Regression Forecast (PHSF2) in Thousands of Units 300 250 200 150 100 50 PHS PHSF1 PHSF2 Nov-98 Jul-98 Mar-98 Nov-97 Jul-97
Mar-97 Nov-96 Jul-96 Mar-96 Nov-95 Jul-95 Mar-95 Nov-94 Jul-94 Mar-94 Nov-93 Jul-93 Mar-93 Nov-92 Jul-92 Mar-92 Nov-91 Jul-91
Mar-91 Nov-90 Jul-90 Mar-90 0 Regression Diagnostics and Residual Analysis It is important to check the adequacy of the model before it becomes part of the decision making process. Residual plots can be used to check the model assumptions. It is important to study outlying observations to decide whether they should be retained or eliminated. If retained, whether their influence should be reduced in the fitting process or revise the regression function. Time Series Data and the Problem of Serial Correlation In the regression models we assume that the errors i are independent.
In business and economics, many regression applications involve time series data. For such data, the assumption of uncorrelated or independent error terms is often not appropriate. Problems of Serial Correlation If the error terms in the regression model are autocorrelated, the use of ordinary least squares procedures has a number of important consequences MSE underestimate the variance of the error terms The confidence intervals and tests using the t and F distribution are no longer strictly applicable. The standard error of the regression coefficients underestimate the variability of the estimated regression coefficients. Spurious regression can result. First order serial correlation The error term in current period is directly related to the error term in the previous time period.
Let the subscript t represent time, then the simple linear regression model is: yt 0 1 xt t Where t = error at time t t t 1 t = the parameter that measures correlation between adjacent error terms t normally distributed error terms with mean zero and variance 2 Example The effect of positive serial correlation in a simple linear regression model. Misleading forecasts of future y values. Standard error of the estimate, S y.x will underestimate the variability of the ys about the true regression line.
Strong autocorrelation can make two unrelated variables appear to be related. Durbin-Watson Test for Serial Correlation Recall the first-order serial correlation model yt 0 1 xt t t t 1 t The hypothesis to be tested are: H 0 : 0 a : 0 is > 0 since in business and The alternativeH hypothesis economic time series tend to show positive correlation. Durbin-Watson Test for Serial Correlation The Durbin-Watson statistic is defined as n (e t DW t 2
et 1 ) 2 n 2 t e t 1 Where et yt y t the residual for time period t et 1 yt 1 y t 1 the residual for time period t - 1 Durbin-Watson Test for Serial Correlation The auto correlation coefficient can be estimated by the lag 1 residual autocorrelation r1(e) n e e t r1 (e) t 1 t 2 n 2
e t t 1 And it can be shown that DW 2(1 r1 (e)) Durbin-Watson Test for Serial Correlation Since 1 < r1(e) < 1 then 0 < DW < 4 If r1(e) = 0, then DW = 2 (there is no correlation.) If r1(e) > 0, then DW < 2 (positive correlation) If r1(e) < 0, Then DW > 2 (negative correlation) Durbin-Watson Test for Serial Correlation Decision rule:
If DW > U, Do not reject H0. If DW < L, Reject H0 If L DW U, the test is inconclusive. The critical Upper (U) an Lower (L) bound can be found in Durbin-Watson table of your text book. To use this table you need to know The significance level () The number of independent parameters in the model (k), and the sample size (n). Example The Blaisdell Company wished to predict its sales by using industry sales as a predictor variable. The following table gives seasonally adjusted quarterly data on company sales and industry sales for the period 1983-1987. Example Year 1983 1984 1985 1986 1987
Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 t 1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 CompSale 20.96 21.4 21.96 21.52 22.39 22.76 23.48 23.66 24.1 24.01 24.54 24.3 25 25.64 26.36 26.98 27.52 27.78 28.24 28.78 InduSale 127.3 130 132.7 129.4
135 137.1 141.2 142.8 145.5 145.3 148.3 146.4 150.2 153.1 157.3 160.7 164.2 165.6 168.7 171.7 Example Blaisdell Company Example Company Sales ($ millions) 35 30 25 20 15 10 5 0 0 50 100
Industry sales($ millions) 150 200 Example The scatter plot suggests that a linear regression model is appropriate. Least squares method was used to fit a regression line to the data. The residuals were plotted against the fitted values. The plot shows that the residuals are consistently above or below the fitted value for extended periods. Example Example To confirm this graphic diagnosis we will use the DurbinWatson test for: H 0 : 0 Ha : 0
The test statistic is: n (e t DW t 2 et 1 ) 2 n 2 t e t 1 Example Year 1983 1984 1985 1986 1987 Quarter 1 2
3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17 18 19 20 Company sales(y) Industry sales(x) 20.96 127.3 21.4 130 21.96 132.7 21.52 129.4 22.39 135 22.76 137.1 23.48 141.2 23.66 142.8 24.1 145.5 24.01 145.3 24.54 148.3 24.3 146.4 25 150.2 25.64 153.1 26.36 157.3
26.98 160.7 27.52 164.2 27.78 165.6 28.24 168.7 28.78 171.7 Blaisdell Company Example ($ 35 30 et -0.02605 -0.06202 0.022021 0.163754 0.04657 0.046377 0.043617 -0.05844 -0.0944 -0.14914 -0.14799 -0.05305 -0.02293 0.105852 0.085464 0.106102 0.029112 0.042316
-0.04416 -0.03301 et -et-1 (et -et-1)^2 -0.03596 0.084036 0.141733 -0.11718 -0.00019 -0.00276 -0.10205 -0.03596 -0.05474 0.001152 0.094937 0.030125 0.12878 -0.02039 0.020638 -0.07699 0.013204 -0.08648 0.011152 0.001293 0.007062 0.020088 0.013732 3.76E-08 7.61E-06 0.010415 0.001293 0.002997
1.33E-06 0.009013 0.000908 0.016584 0.000416 0.000426 0.005927 0.000174 0.007478 0.000124 et ^2 0.000679 0.003846 0.000485 0.026815 0.002169 0.002151 0.001902 0.003415 0.008911 0.022243 0.021901 0.002815 0.000526 0.011205 0.007304 0.011258 0.000848 0.001791 0.00195 0.00109 0.097941 0.133302 Example
.09794 DW .735 .13330 Using Durbin Watson table of your text book, for k = 1, and n=20, and using = .01 we find U = 1.15, and L = .95 Since DW = .735 falls below L = .95 , we reject the null hypothesis, namely, that the error terms are positively autocorrelated. Remedial Measures for Serial Correlation Addition of one or more independent variables to the regression model. One major cause of autocorrelated error terms is the omission from the model of one or more key variables that have time-ordered effects on the dependent variable. Use transformed variables. The regression model is specified in terms of changes rather than levels. Extensions of the Multiple Regression
Model In some situations, nonlinear terms may be needed as independent variables in a regression analysis. Business or economic logic may suggest that nonlinearity is expected. A graphic display of the data may be helpful in determining whether non-linearity is present. One common economic cause for non-linearity is diminishing returns. Fore example, the effect of advertising on sales may diminish as increased advertising is used. Extensions of the Multiple Regression Model Some common forms of nonlinear functions are : Y 0 1 ( X ) 2 ( X 2 ) Y 0 1 ( X ) 2 ( X 2 ) 3 ( X 3 ) Y 0 1 (1 X ) Y e 0 X 1 Extensions of the Multiple Regression
Model To illustrate the use and interpretation of a non-linear term, we return to the problem of developing a forecasting model for private housing starts (PHS). So far we have looked at the following model PHS 0 1 ( MR) 2 (Q 2) 3 (Q3) 4 (Q 4) Where MR is the mortgage rate and Q2, Q3, and Q4 are indicators variables for quarters 2, 3, and 4. Example: Private Housing Start First we add real disposable personal income per capita (DPI) as an independent variable. Our new model for this data set is: PHS 0 1 ( MR) 2 (Q 2) 3 (Q3) 4 (Q 4) 5 ( DPI ) Regression results for this model are shown in the next slide. Example: Private Housing Start SUMMARY OUTPUT Regression Statistics Multiple R 0.943791346 R Square
0.890742104 Adjusted R Square 0.874187878 Standard Error 19.05542121 Observations 39 ANOVA df Regression Residual Total Intercept MR Q2 Q3 Q4 DPI 5 33 38 SS MS F Significance F 97690.01942 19538 53.80753 6.51194E-15 11982.59955 363.1091 109672.619 Coefficients Standard Error -31.06403714
105.1938477 -20.1992545 4.124906847 97.03478074 8.900711541 75.40017073 8.827185877 20.35306822 8.83373887 0.022407799 0.004356973 t Stat -0.2953 -4.8969 10.90191 8.541813 2.304015 5.142974 P-value 0.769613 2.5E-05 1.78E-12 7.17E-10 0.027657 1.21E-05 Lower 95% Upper 95% -245.0826992 182.9546249 -28.59144723 -11.80706176 78.9261326 115.1434289 57.44111179 93.35922967 2.380677107 38.32545934 0.013543464 0.031272134
Example: Private Housing Start The prediction model is PHS 31.06 20.19( MR) 97.03(Q 2) 75.40(Q3) 20.35(Q 4) 0.02( DPI ) In comparison with the previous model, we see that the R-squared has improved.It has changed from 78% to 89%. The standard error of the estimate has decreased from 26.49 for the previous model to 19.05 for the new model. Example: Private Housing Start The value of the DW test has changed from 0.88 for the previous model to 0.78 for the new model. At 5% level the critical value for DW test, from Durbin-Watson table, for k = 5, and n = 39 is L= 1.22, and U = 1.79. Since The value of the DW test is smaller than L=1.22, we reject the null hypothesis H0: =0 This implies that there is serial correlation in both
models, the assumption of the independence of the error terms is not valid. Example: Private Housing Start The Plot of PHS against DPI shows a curve linear relation. Next we introduce a nonlinear term into the regression. The square of disposable personal income per capita (DPI2) is included in the regression model. Private Housing Start and Disposable Personal Income 21500 21000 20500 20000 PHS 19500 19000
18500 18000 17500 0 50 100 150 200 DPI 250 300 350 400 Example: Private Housing Start We also add the dependent variable, lagged one quarter, as an independent variable in order to help reduce serial correlation. The third model that we fit to our data set is:
PHS 0 1 ( MR) 2 (Q 2) 3 (Q3) 4 (Q4) 5 ( DPI ) 6 ( DPI 2 ) 7 ( LPHS ) Regression results for this model are shown in the next slide. Example: Private Housing Start SUMMARY OUTPUT Regression Statistics Multiple R 0.97778626 R Square 0.956065971 Adjusted R Square 0.946145384 Standard Error 12.46719572 Observations 39 ANOVA df Regression Residual Total Intercept MR Q2 Q3 Q4 DPI DPI SQUARED LPHS 7 31
38 SS 104854.2589 4818.360042 109672.619 MS F Significance F 14979.17985 96.37191 3.07085E-19 155.4309691 Coefficients Standard Error t Stat P-value 716.5926532 1017.664989 0.704153784 0.486593 -13.65521724 3.093504134 -4.414158396 0.000114 106.9813297 6.069780998 17.62523718 1.04E-17 27.72122303 9.111432565 3.042465916 0.004748 -13.37855186 7.653050858 -1.748133144 0.09034 -0.060399279 0.104412354 -0.578468704 0.567127 0.000335974 0.000536397 0.626354647 0.535668 0.655786939 0.097265424 6.742241114 1.51E-07 Lower 95% Upper 95%
-1358.949934 2792.13524 -19.96446404 -7.345970448 94.60192287 119.3607366 9.138323433 46.30412262 -28.98706069 2.22995698 -0.273349798 0.15255124 -0.000758014 0.001429963 0.457412689 0.854161189 Example: Private Housing Start The inclusion of DPI2 and Lagged PHS has increased the R-squared to 96% The standard error of the estimate has decreased to 12.45 The value of the DW test has increased to 2.32 which is greater than U = 1.79 which rule out positive serial correlation. You see that the third model worked best for this data set. The following slide gives the data set. Example: Private Housing Start PERIOD
30-Jun-90 30-Sep-90 31-Dec-90 31-Mar-91 30-Jun-91 30-Sep-91 31-Dec-91 31-Mar-92 30-Jun-92 30-Sep-92 31-Dec-92 31-Mar-93 30-Jun-93 30-Sep-93 31-Dec-93 31-Mar-94 30-Jun-94 30-Sep-94 31-Dec-94 31-Mar-95 30-Jun-95 30-Sep-95 31-Dec-95 31-Mar-96 30-Jun-96 30-Sep-96 31-Dec-96 31-Mar-97 30-Jun-97 30-Sep-97 31-Dec-97 31-Mar-98 30-Jun-98 30-Sep-98 31-Dec-98 31-Mar-99
30-Jun-99 30-Sep-99 31-Dec-99 PHS 271.3 233 173.6 146.7 254.1 239.8 199.8 218.5 296.4 276.4 238.8 213.2 323.7 309.3 279.4 252.6 354.2 325.7 265.9 214.2 296.7 308.2 257.2 240 344.5 324 252.4 237.8 324.5 314.6 256.8
258.4 360.4 348 304.6 294.1 377.1 355.6 308.1 MR 10.3372 10.1033 9.9547 9.5008 9.5265 9.2755 8.6882 8.7098 8.6782 8.0085 8.2052 7.7332 7.4515 7.0778 7.0537 7.2958 8.4370 8.5882 9.0977 8.8123 7.9470 7.7012 7.3508 7.2430 8.1050 8.1590
7.7102 7.7905 7.9255 7.4692 7.1980 7.0547 7.0938 6.8657 6.7633 6.8805 7.2037 7.7990 7.8338 LPHS 217 271.3 233 173.6 146.7 254.1 239.8 199.8 218.5 296.4 276.4 238.8 213.2 323.7 309.3 279.4 252.6 354.2 325.7 265.9 214.2
296.7 308.2 257.2 240 344.5 324 252.4 237.8 324.5 314.6 256.8 258.4 360.4 348 304.6 294.1 377.1 355.6 Q2 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 Q3 0 1 0 0 0 1 0 0 0 1 0
0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 Q4 0 0 1 0 0 0
1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 DPI 18063
18031 17856 17748 17861 17816 17811 18000 18085 18036 18330 17975 18247 18246 18413 18154 18409 18493 18667 18834 18798 18871 18942 19071 19081 19161 19152 19331 19315 19385 19478 19632 19719 19905 20194 20377 20472
20756 21124 DPI SQUARED 1,631,359.85 1,625,584.81 1,594,183.68 1,574,957.52 1,595,076.61 1,587,049.28 1,586,158.61 1,620,000.00 1,635,336.13 1,626,486.48 1,679,944.50 1,615,503.13 1,664,765.05 1,664,582.58 1,695,192.85 1,647,838.58 1,694,456.41 1,709,955.25 1,742,284.45 1,773,597.78 1,766,824.02 1,780,573.21 1,793,996.82 1,818,515.21 1,820,422.81 1,835,719.61 1,833,995.52 1,868,437.81 1,865,346.13 1,878,891.13 1,896,962.42 1,927,077.12
1,944,194.81 1,980,963.41 2,038,980.00 2,076,010.87 2,095,440.74 2,153,982.23 2,231,020.37
Expert Group Meeting on Statistical Methodology for Delineating
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