5 Right-Hand Side or Explanatory Variables C = Constant (intercept of the regression line)AYDP92 = Real Disposable Personal Income Growth
6 Estimated Coefficients for Right-Hand Side or Explanatory Variables AYDP92 = (slope of line)Estimated Based on Minimizing the Sum of Squared Error Criterion Under the Assumption that the error averages to zero and has constant variance.
7 Standard Error of the Estimated Coefficients Represent the Likely Sampling Variability (and Consequently, Reliability of the Estimate).
8 The t-stat is the estimated coefficient divided by the standard error. T-StatisticsEach t-statistic provides a test of the hypothesis that the variable is zero or irrelevant (that is, contributes nothing to explaining the dependent variable).The t-stat is the estimated coefficient divided by the standard error.A t-stat whose absolute value is greater than 2 suggests that the explanatory variable is statistically different from zero (that is, it is relevant) at a 95% confidence interval.
9 Probability Statistics or the “p-value” Associated with each coefficient is a probability value, which is the probability of getting a t-statistic at least as large as in absolute value as the one actually obtained, assuming the coefficient is not statistically different from zero.If the t-stat were 2, then the p-value would be The smaller the probability value, the stronger the evidence against an irrelevant variable. Values less than 0.1 are considered a strong evidence against an irrelevant variable.
10 R-SquaredR2 is the percent of the variance of the dependent variable explained by the variables included in the regression.where e is the error and y-bar = the mean of the dependent variable. R-squared ranges between zero and one.
11 R-Squared AdjustedThe Adjusted R-squared is interpreted the same as R-squared but the formula incorporates adjustment for degrees of freedom used in estimating the model.As long as there is more than one right-hand-side variable (including the constant), Adjusted R-squared will be less than R-squared.
12 Standard Error of Regression where T is the number of observations and k is the degrees of freedom (number variables).This is a measure of dispersion and often is compared to the mean of the dependent variable. The smaller the standard error is relative to the mean of the dependent variable the better (and the higher the R-squared).
13 Sum of Squared Residuals This is the minimized value of the sum of squared residuals, which is the objective of least squares. This statistic feeds into other diagnostic measures and in isolation is not very useful.
14 Log LikelihoodThis figure is not used directly. However, an alternative estimation strategy is to maximize the likelihood function to find parameter estimates. For normally distributed errors, maximizing the likelihood function is equivalent to the least square estimates derived from minimizing sum of the squared error.
15 Durbin-Watson Statistic The D-W stat tests for correlation over time in the errors. If errors are not correlated, they are not forecastable. However, when errors are correlated than the overall forecast can be improved by forecasting those errors.If no first-order serial correlation in the errors exist, then the D-W stat = Roughly, a D-W stat of less than 1.5 implies evidence of positive serial correction and over 2.5 implies negative serial correction.
16 Akaike Information Criterion (AIC) Statistic Schwarz Information Criterion (SIC)Both measures penalize for the degrees of freedom, the SIC more harshly than the AIC.These measures often are used to select among competing forecast models (which might include comparison of different lag structures). The lower the AIC or SIC the better relative to a competing model.
17 F-Statistic and Associated Probability This measure tests the hypothesis that the coefficients of all variables in the regression except the constant or intercept term are jointly zero. The associated probability gives the significance level at which we can just reject the hypothesis that all right-hand variables have no predictive power. A reading close to zero allows for rejection of hypothesis (accept the implication that there is explanatory power).
19 EXCEL: Linear Regression “Tools” menu, “Data Analysis”, “Regression”
20 The EXCEL Regression Output Although the Battery of Statistics are More Limited Than from EVIEWS, Those Not Shown in EXCEL Can be Calculated.
21 The EXCEL Regression Output Multiple R (not shown in EVIEWS Output) is the absolute value of the (Pearson) correlation between the observed y values and the predicted (estimated) y values.
22 The EXCEL Regression Output “SS” is the sum of squares.Total = sum of the squared deviations of the y-value from the its mean.Residual = sum of the squared residuals (error) from the regression line. Object of OLS is to minimize this number.Regression = Total - Residual. This represent amount of variance “explained.”
23 The EXCEL Regression Output Sum of Squares“Regression” divided by “Total” gives you R2. For example, (45.33/202.16)=0.224.Also, the “Total” divided by the degrees of freedom(df) is the variance of the y-variable. For example, (202.16/41) = The square root of that is the standard deviation of the y-variable (dependent variable) = 2.22 (percentage points in this example).The EXCEL Regression Output
24 The EXCEL Regression Output “MS” Mean Squares“SS” divided by “df”and“F-Statistic” is the ratio of the “MS Regression” cell and the “MS Residual” cell. For example, (45.33/3.92) =
25 The Five Assumptions Underlying the Classical Linear or Ordinary Least-Squares (OLS) Regression Model for the Bivariate (y = a+bx) and the Multivariate (y= a+bx1+cx2+dx3+…+nxn) Cases
26 Violations of Linearity: Specification Errors Assumption 1: The Dependent Variable Can be Calculated as a Linear Function of the Set of Independent Variables, Plus the Error Term.Violations of Linearity: Specification ErrorsWrong Regressors -- The omission of relevant independent variables or the inclusion of irrelevant independent variables.
27 Violations of Assumption 1: Nonlinearity -- The relationship between the dependent and independent variables is not linear.Changing Parameters -- The estimated parameters (the coefficients) do not remain constant for the period included in the regression.
28 Assumption 2: The Expected Value (Mean) of the Error Term is Zero, that is, the Mean of the Distribution from which the Sample is Drawn is Zero.Violation of Zero MeanBiased Intercept -- If the error term is systematically positive or negative, you can think of rearranging the terms to re-establish a zero mean by defining a “new error term” (old error term = new error term + bias). Then the new constant term will equal old constant term + bias.
29 Violation of Uncorrelated Constant Variance Assumption 3: The Error Terms have the Same Variance (homoskedasticity) and are not correlated with each other.Violation of Uncorrelated Constant VarianceHeteroskedasticity -- The error terms do not have constant variance.Autocorrelated Errors -- The error terms are correlated with each other.
30 Violations of Assumption 4 Assumption 4: The Independent Variables can be Considered Fixed in Repeated Sampling.Violations of Assumption 4Errors in Variables -- Measurement error in the data.Autoregression -- Using lagged values of the independent variable.Simultaneous Equation Estimation -- The dependent variables are simultaneously determined by the interaction of other relationships.
31 Violation of Assumption 5 Assumption 5: The Number of Observations is Greater than the Number of Independent Variables and there is No Exact Linear Relationship Between the Independent Variables.Violation of Assumption 5Multicollinearity -- Two or more independent variables are approximately linearly related in the sample period.
32 Popularity of OLS 1. Computational Convenience. 2. Ease of Interpretation.3. The OLS estimates of the coefficients will the BEST LINEAR UNBIASED ESTIMATES (BLUE) of the true coefficients.
33 Extensions of OLS are generally to Handle Violations of These Assumptions
34 Once You have Estimated Your Equations and Developed Your Model, Then the Next Step is to Simulate the Model.
35 A System of Equations Can Be Solved Using the Gauss-Seidel Method You Do Not Need to Know too Much About This Since it is Built Into Either EXCEL (if you turn the option on) or EViews.
36 Forecast Simulation Topics The Gauss-Seidel Method.EXCEL Simulation Model.
37 Look at GAUSSSEI.XLS Which is Found on Website Demonstrates the Concept of Solving a Set of Simulataneous Equations.