EXCEL 2007: Multiple Regression

A. Colin Cameron, Dept. of Economics, Univ. of Calif. - Davis

  • This January 2009 help sheet gives information on
  • There is little extra to know beyond regression with one explanatory variable.
    The main addition is the F-test for overall fit.


    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:

    Multiple regression data

    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.

    Multiple 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

    Multiple regression

    The regression output has three components:


    This is the following output. Of greatest interest is R Square.

    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.


    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 ypredicted 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.


    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).


    A simple summary of the above output is that the fitted line is

         y = 0.8966 + 0.3365*x + 0.0021*z


    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%.


    The coefficient of HH SIZE has estimated standard error of 0.4227, t-statistic of 0.7960 and p-value 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.


    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.

      t = (b2 - H0 value of β2) / (standard error of b2 )
        = (0.33647 - 1.0) / 0.42270
        = -1.569.

    Using the p-value approach

    Using the critical value approach


    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.


    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 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