EXCEL 2007: Multiple Regression A. Colin Cameron, Dept. of Economics, Univ. of Calif. - Davis This January 2009 help sheet gives information on
Multiple regression using the Data Analysis Add-in. Interpreting the regression statistic. Interpreting the ANOVA table (often this is skipped). Interpreting the regression coefficients table. Confidence intervals for the slope parameters. Testing for statistical significance of coefficients Testing hypothesis on a slope parameter. Testing overall significance of the regressors. Predicting y given values of regressors. Excel limitations.
There is little extra to know beyond regression with one explanatory variable. The main addition is the F-test for overall fit.
MULTIPLE REGRESSION USING THE DATA ANALYSIS ADD-IN This requires the Data Analysis Add-in: see Excel 2007: Access and Activating the Data Analysis Add-in The data used are in carsdata.xls We then create a new variable in cells C2:C6, cubed household size as a regressor. Then in cell C1 give the the heading CUBED HH SIZE. (It turns out that for the se data squared HH SIZE has a coefficient of exactly 0.0 the cube is used). The spreadsheet cells A1:C6 should look like:
We have regression with an intercept and the regressors HH SIZE and CUBED HH SIZE
The population regression model is: y = β1 + β2 x2 + β3 x3 + u It is assumed that the error u is independent with constant variance (homoskedastic) - see EXCEL LIMITATIONS at the bottom. We wish to estimate the regression line:
y = b1 + b2 x2 + b3 x3
We do this using the Data analysis Add-in and Regression.
The only change over one-variable regression is to include more than one column in the Input X Range. Note, however, that the regressors need to be in contiguous columns (here columns B and C). If this is not the case in the original data, then columns need to be copied to get the regressors in contiguous columns.
Hitting OK we obtain
The regression output has three components:
Regression statistics table ANOVA table Regression coefficients table.
INTERPRET REGRESSION STATISTICS TABLE This is the following output. Of greatest interest is R Square.
Explanation Multiple R
0.895828 R = square root of R2
R Square
0.802508 R2
Adjusted R Square 0.605016 Adjusted R2 used if more than one x variable Standard Error
0.444401 This is the sample estimate of the standard deviation of the error u
Observations
5
Number of observations used in the regression (n)
The above gives the overall goodness-of-fit measures: R2 = 0.8025
Correlation between y and y-hat is 0.8958 (when squared gives 0.8025). Adjusted R2 = R2 - (1-R2 )*(k-1)/(n-k) = .8025 - .1975*2/2 = 0.6050. The standard error here refers to the estimated standard deviation of the error term u. It is sometimes called the standard error of the regression. It equals sqrt(SSE/(n-k)). It is not to be confused with the standard error of y itself (from descriptive statistics) or with the standard errors of the regression coefficients given below. R2 = 0.8025 means that 80.25% of the variation of yi around ybar (its mean) is explained by the regressors x2i and x3i.
INTERPRET ANOVA TABLE An ANOVA table is given. This is often skipped.
df SS
MS
F
Significance F
Regression 2 1.6050 0.8025 4.0635 0.1975 Residual
2 0.3950 0.1975
Total
4 2.0
The ANOVA (analysis of variance) table splits the sum of squares into its components. Total sums of squares = Residual (or error) sum of squares + Regression (or explained) sum of squares. Thus Σ i (yi - ybar)2 = Σ i (yi - yhati)2 + Σ i (yhati - ybar)2 where yhati is the value of yi predicted from the regression line and ybar is the sample mean of y. For example: R2 = 1 - Residual SS / Total SS (general formula for R2) = 1 - 0.3950 / 1.6050 (from data in the ANOVA table) = 0.8025 (which equals R2 given in the regression Statistics table). The column labeled F gives the overall F-test of H0: β2 = 0 and β3 = 0 versus Ha: at least one of β2 and β3 does not equal zero. Aside: Excel computes F this as: F = [Regression SS/(k-1)] / [Residual SS/(n-k)] = [1.6050/2] / [.39498/2] = 4.0635. The column labeled significance F has the associated P-value. Since 0.1975 > 0.05, we do not reject H0 at signficance level 0.05. Note: Significance F in general = FINV(F, k-1, n-k) where k is the number of regressors
including hte intercept. Here FINV(4.0635,2,2) = 0.1975. INTERPRET REGRESSION COEFFICIENTS TABLE The regression output of most interest is the following table of coefficients and associated output:
Coefficient St. error t Stat
P-value Lower 95% Upper 95%
Intercept
0.89655
0.76440 1.1729 0.3616 -2.3924
4.1855
HH SIZE
0.33647
0.42270 0.7960 0.5095 -1.4823
2.1552
CUBED HH SIZE 0.00209
0.01311 0.1594 0.8880 -0.0543
0.0585
Let βj denote the population coefficient of the jth regressor (intercept, HH SIZE and CUBED HH SIZE). Then
Column "Coefficient" gives the least squares estimates of βj. Column "Standard error" gives the standard errors (i.e.the estimated standard deviation) of the least squares estimates bj of βj. Column "t Stat" gives the computed t-statistic for H0: βj = 0 against Ha: βj ≠ 0. This is the coefficient divided by the standard error. It is compared to a t with (n-k) degrees of freedom where here n = 5 and k = 3.
Column "P-value" gives the p-value for test of H0: βj = 0 against Ha: βj ≠ 0.. This equals the Pr{|t| > t-Stat}where t is a t-distributed random variable with n-k degrees of freedom and t-Stat is the computed value of the t-statistic given in the previous column. Note that this p-value is for a two-sided test. For a one-sided test divide this p-value by 2 (also checking the sign of the t-Stat).
Columns "Lower 95%" and "Upper 95%" values define a 95% confidence interval for βj.
A simple summary of the above output is that the fitted line is y = 0.8966 + 0.3365*x + 0.0021*z
CONFIDENCE INTERVALS FOR SLOPE COEFFICIENTS 95% confidence interval for slope coefficient β2 is from Excel output (-1.4823, 2.1552).
Excel computes this as b2 ± t_.025(3) × se(b2) = 0.33647 ± TINV(0.05, 2) × 0.42270 = 0.33647 ± 4.303 × 0.42270 = 0.33647 ± 1.8189 = (-1.4823, 2.1552). Other confidence intervals can be obtained. For example, to find 99% confidence intervals: in the Regression dialog box (in the Data Analysis Add-in), check the Confidence Level box and set the level to 99%. TEST HYPOTHESIS OF ZERO SLOPE COEFFICIENT ("TEST OF STATISTICAL SIGNIFICANCE") The coefficient of HH SIZE has estimated standard error of 0.4227, t-statistic of 0.7960 and pvalue of 0.5095. It is therefore statistically insignificant at significance level α = .05 as p > 0.05. The coefficient of CUBED HH SIZE has estimated standard error of 0.0131, t-statistic of 0.1594 and p-value of 0.8880. It is therefore statistically insignificant at significance level α = .05 as p > 0.05. There are 5 observations and 3 regressors (intercept and x) so we use t(5-3)=t(2). For example, for HH SIZE p = =TDIST(0.796,2,2) = 0.5095.
TEST HYPOTHESIS ON A REGRESSION PARAMETER Here we test whether HH SIZE has coefficient β2 = 1.0. Example: H0: β2 = 1.0 against Ha: β2 ≠ 1.0 at significance level α = .05. Then t = (b2 - H0 value of β2) / (standard error of b2 ) = (0.33647 - 1.0) / 0.42270 = -1.569. Using the p-value approach
p-value = TDIST(1.569, 2, 2) = 0.257. [Here n=5 and k=3 so n-k=2]. Do not reject the null hypothesis at level .05 since the p-value is > 0.05.
Using the critical value approach
We computed t = -1.569
The critical value is t_.025(2) = TINV(0.05,2) = 4.303. [Here n=5 and k=3 so n-k=2]. So do not reject null hypothesis at level .05 since t = |-1.569| < 4.303.
OVERALL TEST OF SIGNIFICANCE OF THE REGRESSION PARAMETERS We test H0: β2 = 0 and β3 = 0 versus Ha: at least one of β2 and β3 does not equal zero. From the ANOVA table the F-test statistic is 4.0635 with p-value of 0.1975. Since the p-value is not less than 0.05 we do not reject the null hypothesis that the regression parameters are zero at significance level 0.05. Conclude that the parameters are jointly statistically insignificant at significance level 0.05. Note: Significance F in general = FINV(F, k-1, n-k) where k is the number of regressors including hte intercept. Here FINV(4.0635,2,2) = 0.1975.
PREDICTED VALUE OF Y GIVEN REGRESSORS Consider case where x = 4 in which case CUBED HH SIZE = x^3 = 4^3 = 64. yhat = b1 + b2 x2 + b3 x3 = 0.88966 + 0.3365×4 + 0.0021×64 = 2.37006
EXCEL LIMITATIONS Excel restricts the number of regressors (only up to 16 regressors ??). Excel requires that all the regressor variables be in adjoining columns. You may need to move columns to ensure this. e.g. If the regressors are in columns B and D you need to copy at least one of columns B and D so that they are adjacent to each other. Excel standard errors and t-statistics and p-values are based on the assumption that the error is independent with constant variance (homoskedastic). Excel does not provide alternaties, such asheteroskedastic-robust or autocorrelation-robust standard errors and t-statistics and p-values. More specialized software such as STATA, EVIEWS, SAS, LIMDEP, PC-TSP, ... is needed. For further information on how to use Excel go to http://cameron.econ.ucdavis.edu/excel/excel.html
EXCEL 2007: Multiple Regression
A. Colin Cameron, Dept. of Economics, Univ. of Calif. - Davis This January 2009 help sheet gives information on
Multiple regression using the Data Analysis Add-in. Interpreting the regression statistic. Interpreting the ANOVA table (often this is skipped). Interpreting the regression coefficients table. Confidence intervals for the slope parameters. Testing for statistical significance of coefficients Testing hypothesis on a slope parameter. Testing overall significance of the regressors. Predicting y given values of regressors. Excel limitations.
There is little extra to know beyond regression with one explanatory variable. The main addition is the F-test for overall fit.
MULTIPLE REGRESSION USING THE DATA ANALYSIS ADD-IN This requires the Data Analysis Add-in: see Excel 2007: Access and Activating the Data Analysis Add-in The data used are in carsdata.xls We then create a new variable in cells C2:C6, cubed household size as a regressor. Then in cell C1 give the the heading CUBED HH SIZE. (It turns out that for the se data squared HH SIZE has a coefficient of exactly 0.0 the cube is used). The spreadsheet cells A1:C6 should look like:
We have regression with an intercept and the regressors HH SIZE and CUBED HH SIZE The population regression model is: y = β1 + β2 x2 + β3 x3 + u It is assumed that the error u is independent with constant variance (homoskedastic) - see EXCEL LIMITATIONS at the bottom.
We wish to estimate the regression line:
y = b1 + b2 x2 + b3 x3
We do this using the Data analysis Add-in and Regression.
The only change over one-variable regression is to include more than one column in the Input X Range. Note, however, that the regressors need to be in contiguous columns (here columns B and C). If this is not the case in the original data, then columns need to be copied to get the regressors in contiguous columns. Hitting OK we obtain
The regression output has three components:
Regression statistics table ANOVA table Regression coefficients table.
INTERPRET REGRESSION STATISTICS TABLE This is the following output. Of greatest interest is R Square.
Explanation Multiple R
0.895828 R = square root of R2
R Square
0.802508 R2
Adjusted R Square 0.605016 Adjusted R2 used if more than one x variable Standard Error
0.444401 This is the sample estimate of the standard deviation of the error u
Observations
5
Number of observations used in the regression (n)
The above gives the overall goodness-of-fit measures: R2 = 0.8025 Correlation between y and y-hat is 0.8958 (when squared gives 0.8025). Adjusted R2 = R2 - (1-R2 )*(k-1)/(n-k) = .8025 - .1975*2/2 = 0.6050.
The standard error here refers to the estimated standard deviation of the error term u. It is sometimes called the standard error of the regression. It equals sqrt(SSE/(n-k)). It is not to be confused with the standard error of y itself (from descriptive statistics) or with the standard errors of the regression coefficients given below. R2 = 0.8025 means that 80.25% of the variation of yi around ybar (its mean) is explained by the regressors x2i and x3i.
INTERPRET ANOVA TABLE An ANOVA table is given. This is often skipped.
df SS
MS
F
Significance F
Regression 2 1.6050 0.8025 4.0635 0.1975 Residual
2 0.3950 0.1975
Total
4 2.0
The ANOVA (analysis of variance) table splits the sum of squares into its components. Total sums of squares = Residual (or error) sum of squares + Regression (or explained) sum of squares. Thus Σ i (yi - ybar)2 = Σ i (yi - yhati)2 + Σ i (yhati - ybar)2 where yhati is the value of yi predicted from the regression line and ybar is the sample mean of y. For example: R2 = 1 - Residual SS / Total SS (general formula for R2) = 1 - 0.3950 / 1.6050 (from data in the ANOVA table) = 0.8025 (which equals R2 given in the regression Statistics table). The column labeled F gives the overall F-test of H0: β2 = 0 and β3 = 0 versus Ha: at least one of β2 and β3 does not equal zero. Aside: Excel computes F this as: F = [Regression SS/(k-1)] / [Residual SS/(n-k)] = [1.6050/2] / [.39498/2] = 4.0635. The column labeled significance F has the associated P-value. Since 0.1975 > 0.05, we do not reject H0 at signficance level 0.05. Note: Significance F in general = FINV(F, k-1, n-k) where k is the number of regressors including hte intercept. Here FINV(4.0635,2,2) = 0.1975.
INTERPRET REGRESSION COEFFICIENTS TABLE The regression output of most interest is the following table of coefficients and associated output:
Coefficient St. error t Stat
P-value Lower 95% Upper 95%
Intercept
0.89655
0.76440 1.1729 0.3616 -2.3924
4.1855
HH SIZE
0.33647
0.42270 0.7960 0.5095 -1.4823
2.1552
CUBED HH SIZE 0.00209
0.01311 0.1594 0.8880 -0.0543
0.0585
Let βj denote the population coefficient of the jth regressor (intercept, HH SIZE and CUBED HH SIZE). Then
Column "Coefficient" gives the least squares estimates of βj. Column "Standard error" gives the standard errors (i.e.the estimated standard deviation) of the least squares estimates bj of βj. Column "t Stat" gives the computed t-statistic for H0: βj = 0 against Ha: βj ≠ 0. This is the coefficient divided by the standard error. It is compared to a t with (n-k) degrees of freedom where here n = 5 and k = 3.
Column "P-value" gives the p-value for test of H0: βj = 0 against Ha: βj ≠ 0.. This equals the Pr{|t| > t-Stat}where t is a t-distributed random variable with n-k degrees of freedom and t-Stat is the computed value of the t-statistic given in the previous column. Note that this p-value is for a two-sided test. For a one-sided test divide this p-value by 2 (also checking the sign of the t-Stat).
Columns "Lower 95%" and "Upper 95%" values define a 95% confidence interval for βj.
A simple summary of the above output is that the fitted line is y = 0.8966 + 0.3365*x + 0.0021*z
CONFIDENCE INTERVALS FOR SLOPE COEFFICIENTS 95% confidence interval for slope coefficient β2 is from Excel output (-1.4823, 2.1552). Excel computes this as b2 ± t_.025(3) × se(b2)
= 0.33647 ± TINV(0.05, 2) × 0.42270 = 0.33647 ± 4.303 × 0.42270 = 0.33647 ± 1.8189 = (-1.4823, 2.1552). Other confidence intervals can be obtained. For example, to find 99% confidence intervals: in the Regression dialog box (in the Data Analysis Add-in), check the Confidence Level box and set the level to 99%. TEST HYPOTHESIS OF ZERO SLOPE COEFFICIENT ("TEST OF STATISTICAL SIGNIFICANCE") The coefficient of HH SIZE has estimated standard error of 0.4227, t-statistic of 0.7960 and pvalue of 0.5095. It is therefore statistically insignificant at significance level α = .05 as p > 0.05. The coefficient of CUBED HH SIZE has estimated standard error of 0.0131, t-statistic of 0.1594 and p-value of 0.8880. It is therefore statistically insignificant at significance level α = .05 as p > 0.05. There are 5 observations and 3 regressors (intercept and x) so we use t(5-3)=t(2). For example, for HH SIZE p = =TDIST(0.796,2,2) = 0.5095.
TEST HYPOTHESIS ON A REGRESSION PARAMETER Here we test whether HH SIZE has coefficient β2 = 1.0. Example: H0: β2 = 1.0 against Ha: β2 ≠ 1.0 at significance level α = .05. Then t = (b2 - H0 value of β2) / (standard error of b2 ) = (0.33647 - 1.0) / 0.42270 = -1.569. Using the p-value approach
p-value = TDIST(1.569, 2, 2) = 0.257. [Here n=5 and k=3 so n-k=2]. Do not reject the null hypothesis at level .05 since the p-value is > 0.05.
Using the critical value approach
We computed t = -1.569 The critical value is t_.025(2) = TINV(0.05,2) = 4.303. [Here n=5 and k=3 so n-k=2]. So do not reject null hypothesis at level .05 since t = |-1.569| < 4.303.
OVERALL TEST OF SIGNIFICANCE OF THE REGRESSION PARAMETERS We test H0: β2 = 0 and β3 = 0 versus Ha: at least one of β2 and β3 does not equal zero. From the ANOVA table the F-test statistic is 4.0635 with p-value of 0.1975. Since the p-value is not less than 0.05 we do not reject the null hypothesis that the regression parameters are zero at significance level 0.05. Conclude that the parameters are jointly statistically insignificant at significance level 0.05. Note: Significance F in general = FINV(F, k-1, n-k) where k is the number of regressors including hte intercept. Here FINV(4.0635,2,2) = 0.1975.
PREDICTED VALUE OF Y GIVEN REGRESSORS Consider case where x = 4 in which case CUBED HH SIZE = x^3 = 4^3 = 64. yhat = b1 + b2 x2 + b3 x3 = 0.88966 + 0.3365×4 + 0.0021×64 = 2.37006
EXCEL LIMITATIONS Excel restricts the number of regressors (only up to 16 regressors ??). Excel requires that all the regressor variables be in adjoining columns. You may need to move columns to ensure this. e.g. If the regressors are in columns B and D you need to copy at least one of columns B and D so that they are adjacent to each other. Excel standard errors and t-statistics and p-values are based on the assumption that the error is independent with constant variance (homoskedastic). Excel does not provide alternaties, such asheteroskedastic-robust or autocorrelation-robust standard errors and t-statistics and p-values. More specialized software such as STATA, EVIEWS, SAS, LIMDEP, PC-TSP, ... is needed. For further information on how to use Excel go to http://cameron.econ.ucdavis.edu/excel/excel.html
