Due:Thursday, September 18, by noon

Data: ps_1_data.xls (Excel Spreadsheet)
Solution: ps_1_solution.xls (Excel Spreadsheet)

Introduction: This problem set is based on material in Chapter 6. This problem set is designed to familiarize you with the linear regression feature found in spreadsheets as well as concepts from Chapter 6 of the text.

Instructions: Open the spreadsheet ps_1_data.xls in Excel. This spreadsheet contains the following variables:

• C_i: Consumption Expenditires in Solvia(Column A)
• Y_i: National Income in Solvia (Column B)

Problem Set Question: Solvia is a closed primitive economy. It does not trade with foreign countries and it does not not have a government. The total product or total income in Solvia is used for consumption and investment

1. Use the OLS method to estimate all parameters of the following empirical models
   Model 1: C_i = a₁ + b₁Y_i + e_1i
   Model 2: C_i = a₂ + b₂Z_i + e_2i

   Where Z=Y-C so Z is the value of autonomous investment expenditure in Solvovia.

   Model 1

   SUMMARY OUTPUT				
   				
   Regression Statistics				
   Multiple R        0.98
   R Square          0.96
   Adjusted R Square 0.96
   Standard Error    3.39
   Observations        12			
   				
   ANOVA
               df      SS        MS     F
   Regression   1   3568.73   3568.73 309.60
   Residual    10    115.26     11.52	
   Total       11   3684		
   
                Coefficients    Standard Error    t Stat	
   Intercept           2.129             7.164     0.297
   X Variable 1        0.861             0.048    17.596
   

   
   Model 2

   Regression Statistics				
   Multiple R          0.53			
   R Square            0.28			
   Adjusted R Square   0.20			
   Standard Error     16.33			
   Observations          12			
   				
   ANOVA				
               df     SS       MS     F
   Regression   1  1017.31  1017.31   3.81
   Residual    10  2666.69   266.67	
   Total       11  3684		
   				
              Coefficients Standard Error   t Stat
   Intercept        87.192         20.919    4.168
   X Variable 1      2.212          1.132    1.953
   

2. Explain what the coeffecient estimates from the regression model mean.

   For Model 1, the intercept or constant is the amount of consumption spending that takes place when income is zero. The parameter on variable 1 is themarginal propensity to consume out of income.

   For Model 2, the variable Z_i is investment spending. The model is a linear relationship between consumption and investment, and the parameters are the intercept and slope of this linear relationship.

3. Use a t-test to test the statistical significance of b₁ and b₂

   The details of the procedure are on page 211 (2. Testing of Hypotheses). Excel automatically generates the two tailed t-test.
   Test for parameter b₁
   

   • a. Set null and alternative hypotheses.
     H₀: b₁ = 0
     H₁: b₁ not equal 0
     
   • b. Calculate t-statistic.
     T-statistic: 17.596
     
   • c. Compare the computed t-value to the critical t-value from a table of t-sistributions.
     To find the 5% critical value for a two tailed test, use the Excel function: TINV(0.025,11)
     17.596 > 2.593
     
   • d. If the absolute value of computed t is greater, reject the null, else accept.
     The null hypothesis is rejected. The data are consistent with the assumptuion that the marginal propensity to consume is not zero.

4. Use the regression results from Model 1 to predict the mean value of C when Y=200. Construct a 95% confidence interval for this predicted value.

   Prediction:
   The regression model is
   C = 87.192 + 2.212 x Y
   Plug Y=200 into this equation to get the predicted value of C when Y=200
   C = 87.192 + 2.212 (200) = 174

   Confidence Interval
   Use equation (6.90), page 212 in the text to find the confidence interval.
   The confidence interval is [167,182]. The spreadsheet has the details.

5. Explain each of these statistics from the regressions mean: R², SEE, S_e².

   SEE is the standard error of the estimate. This is a measure of the variation of the residuals from the regression model. See equation (6.77). S_e² is the variance of the resuduals from the regression model. It equals SEE². R² is the coefficient of determination. It shows the fraction of variation in the dependent variable (C) accounted for by variation in the independent variable. See equation (6.81).

6. Explain what information you would need in order to use this model to predict the level of consumption expenditures in Solvia next year.

   Need a forecast of income in Solvia next year. The regression model shows how current consumption spending depends of current income. This implies that future consumption could be predicted using future income.