Regression Output

This is the output from a linear regression with a description of what each element shows. Note that this set of output is specific to least squares regression; other forms of estimation will be similar, but may leave out some statistics that aren’t defined or have no usable interpretation.

(a) Linear Regression - Estimation by Least Squares

(b) Dependent Variable RATE

(d) Usable Observations 443

(e) Degrees of Freedom 439

(f) Centered R^2 0.2526958

(g) R-Bar^2 0.2475890

(h) Uncentered R^2 0.8717009

(i) Mean of Dependent Variable 6.0806546275

(j) Std Error of Dependent Variable 2.7714419161

(k) Standard Error of Estimate 2.4039938631

(l) Sum of Squared Residuals 2537.0628709

(m) Regression F(3,439) 49.4816

(n) Significance Level of F 0.0000000

(o) Log Likelihood -1015.1499

(p) Durbin-Watson Statistic 0.0816

(q)Variable (r) Coeff (s) Std Error (t) T-Stat (u) Signif

*********************************************************************************

1. Constant 2.11841571 0.42030566 5.04018 0.00000068

2. IP 0.06417324 0.00768853 8.34662 0.00000000

3. M1DIFF -0.04183328 0.01485687 -2.81575 0.00508547

4. PPISUM 58.26459284 8.01033322 7.27368 0.00000000

We’ll use the following notation:

y	The vector of values for the dependent variable
\(\bar y\)	The sample mean of the dependent variable over the estimation range
\({\bf{\tilde y}}\)	The deviations from the mean of the dependent variable
\(e\)	The vector of residuals
\(T\)	The number of observations
\(K\)	regressors

(a)	The type of model and estimation technique used.
(b)	The dependent variable
(c)	If you are using a CALENDAR, RATS will list the frequency of the data and the beginning and ending of the estimation range. If you have data without a date scheme, this will be skipped.
(d)	The number of usable entries in the estimation range: \(T\)
(e)	The degrees of freedom: \(T-K\)
(f)	The centered \(R^2\) statistic: \(1 - \frac{{{\bf{e'}}{\kern 1pt} {\bf{e}}}}{{{\bf{\tilde y'}}{\kern 1pt} {\bf{\tilde y}}}}\)
(g)	The adjusted \(R^2\) statistic (\(\bar R^2 \)): \(1 - \left( {\frac{{{\bf{e'e}}/\left( {T - K} \right)}}{{{\bf{\tilde y'\tilde y}}/\left( {T - 1} \right)}}} \right)\)
(h)	The uncentered \(R^2\) statistic: \(1 - \frac{{{\bf{e'}}{\kern 1pt} {\bf{e}}}}{{{\bf{y'y}}}}\)
(i)	The mean of the dependent variable: \(\bar y\)
(j)	The standard error of the dependent variable: \(\sqrt {\frac{{{\bf{\tilde y'}}{\kern 1pt} {\bf{\tilde y}}}}{{\left( {T - 1} \right)}}} \)
(k)	The Standard Error of Estimate: \(\sqrt {\frac{{{\bf{e'}}{\kern 1pt} {\bf{e}}}}{{\left( {T - K} \right)}}} \)
(l)	The Sum of Squared Residuals: \({\bf{e}}'{\bf{e}}\)
(m)	The regression F-statistic: \(\frac{{\frac{{{\bf{\tilde y'}}{\kern 1pt} {\bf{\tilde y}} - {\bf{e'}}{\kern 1pt} {\bf{e}}}}{{\left( {K - 1} \right)}}}}{{\frac{{{\bf{e'}}{\kern 1pt} {\bf{e}}}}{{\left( {T - K} \right)}}}}\)
(n)	The marginal significance level of the \(F\), with \(K-1\) and \(T-K\) degrees of freedom.
(o)	The log-likelihood: \(- \frac{T}{2}\left( {\log \left( {\frac{{{\bf{e'e}}}}{T}} \right) + 1 + \log \left( {2\pi } \right)} \right)\)
(p)	The Durbin-Watson statistic: \(\frac{{\sum\limits_{t = 2}^T {\left( {e_t - e_{t - 1} } \right)^2 } }}{{\sum\limits_{t = 1}^T {e_t^2 } }}\)
(q)	The names of the explanatory variables. Lags are shown as name{lag}.
(r)	The estimated coefficients.
(s)	The standard error of the coefficient estimate.
(t)	The t-statistic of the coefficient (coefficient/its standard error).
(u)	The marginal significance level for a (two-tailed) test for a zero coefficient.

Goodness of Fit Measures

You will notice there are three versions of the \(R^2\) statistic: the centered \(R^2\), the \(\bar R^2 \) (\(R^2\) adjusted for degrees of freedom) and the uncentered \(R^2\). The centered and adjusted \(R^2\) are typically the only ones of interest.

RATS also displays the mean and standard error of the dependent variable. These are simply statistics on the dependent variable, and tell you nothing about the accuracy of the regression model. They are the same values you would get by doing a STATISTICS instruction on the RATE series over the same range.

Regression F and Significance Level

These are only included if the regression includes a CONSTANT (or its equivalent in some set of dummy variables). This is the result of an F-test for the hypothesis that all coefficients (excluding CONSTANT) are zero. The numbers in parentheses after the F are the degrees of freedom for the numerator and denominator, respectively.

Log Likelihood

For a linear regression, this is the log likelihood of the data assuming Normal residuals. Note that this includes the “constants”: the \(1 + \log (2\pi )\) terms only interact with \(T\) and not with the residuals and so could be dropped from any comparison of two linear regressions with the same number of observations. While some programs omit these, RATS always includes them in all calculations of likelihoods or any density functions.

Durbin-Watson Statistic

The Durbin-Watson statistic tests for first-order serial correlation in the residuals. The ideal result is 2.0, indicating the absence of first-order serial correlation. Values lower than 2.0 (and particularly below 1.0) suggest that the residuals may be serially correlated.

RATS always computes a Durbin-Watson statistic, even if you have cross-section data where “serial correlation” isn’t likely to be an issue. However, you should keep in mind that the tabled values for the Durbin-Watson statistic are known to be invalid in a variety of circumstances, such as the presence of lagged dependent variables.

Coefficient Table

RATS attempts to come up with a common format for displaying the regression coefficients and standard errors. This makes the results easier to read than if the decimal points didn’t align. The table lists all the variables in the regression, including the constant if you included it. The Coeff column lists the estimated coefficients for each variable. The t-statistic is the ratio of a coefficient to its standard error.

Signif is the significance level for a two-tailed test for a zero coefficient. For models estimated by least squares, the t-statistic in the T-Stat column is treated as having a t distribution with \(T-K\) degrees of freedom. If the model is estimated by some other technique, the t-statistic is treated as having a Normal distribution.