Introductory Econometrics Using R

Also covered using Python and Stata

library(wooldridge)
library(stargazer)
library(AER)
library(dynlm)

Example 18.1. Housing Investment and Residential Price Inflation

tshseinv <- ts(hseinv)
reg1 <- dynlm(linvpc ~ t, data = tshseinv)
stargazer(reg1, no.space=TRUE, type="text")

## 
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                               linvpc           
## -----------------------------------------------
## t                            0.008***          
##                               (0.002)          
## Constant                     -0.841***         
##                               (0.045)          
## -----------------------------------------------
## Observations                    42             
## R2                             0.335           
## Adjusted R2                    0.319           
## Residual Std. Error       0.142 (df = 40)      
## F Statistic           20.190*** (df = 1; 40)   
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01

u <- resid(reg1)
RationalDL <- dynlm( u ~ gprice + L(u, 1) +  L(gprice, 1), data = tshseinv)
GeometricDL <- dynlm( u ~ gprice + L(u, 1), data = tshseinv)

stargazer(GeometricDL, RationalDL, title="Table 18.1 Distributed Lag Models for Housing Investment: log(invpc)", column.labels=c("GeometricDL", "RationalDL"), no.space=TRUE, type="text")

## 
## Table 18.1 Distributed Lag Models for Housing Investment: log(invpc)
## =================================================================
##                                  Dependent variable:             
##                     ---------------------------------------------
##                                           u                      
##                          GeometricDL             RationalDL      
##                              (1)                    (2)          
## -----------------------------------------------------------------
## gprice                     3.095***               3.256***       
##                            (0.933)                (0.970)        
## L(u, 1)                    0.340**                0.547***       
##                            (0.132)                (0.152)        
## L(gprice, 1)                                     -2.936***       
##                                                   (0.973)        
## Constant                    -0.010                 0.006         
##                            (0.018)                (0.017)        
## -----------------------------------------------------------------
## Observations                  41                     40          
## R2                          0.407                  0.542         
## Adjusted R2                 0.375                  0.504         
## Residual Std. Error    0.111 (df = 38)        0.100 (df = 36)    
## F Statistic         13.022*** (df = 2; 38) 14.200*** (df = 3; 36)
## =================================================================
## Note:                                 *p<0.1; **p<0.05; ***p<0.01

Example 18.2. Unit Root Test for Three-Month T-Bill Rates

tsintqrt <- ts(intqrt)
uroot1 <- dynlm(cr3 ~ L(r3, 1), data = tsintqrt)
rho <- 1 + uroot1$coef[2]
rho

##  L(r3, 1) 
## 0.9092894

uroot2 <- dynlm(r3 ~ L(r3, 1), data = tsintqrt)
stargazer(uroot1, uroot2, no.space=TRUE, type="text")

## 
## ===========================================================
##                                    Dependent variable:     
##                                ----------------------------
##                                     cr3            r3      
##                                     (1)            (2)     
## -----------------------------------------------------------
## L(r3, 1)                          -0.091**      0.909***   
##                                   (0.037)        (0.037)   
## Constant                          0.625**        0.625**   
##                                   (0.261)        (0.261)   
## -----------------------------------------------------------
## Observations                        123            123     
## R2                                 0.048          0.836    
## Adjusted R2                        0.040          0.834    
## Residual Std. Error (df = 121)     1.228          1.228    
## F Statistic (df = 1; 121)         6.116**      614.595***  
## ===========================================================
## Note:                           *p<0.1; **p<0.05; ***p<0.01

Example 18.3. Unit Root Test for Annual U.S. Inflation

tsphillips <- ts(subset(phillips, phillips$year<1997))

uroot3 <- dynlm(cinf ~ inf_1 + L(cinf, 1) , data = tsphillips)
rho1 <- 1 + uroot3$coef[2]
rho1

##     inf_1 
## 0.6896748

reg2 <- dynlm(cinf ~ L(inf,1), data = tsphillips)
rho2 <- 1 + reg2$coef[2]
rho2

## L(inf, 1) 
## 0.6652586

stargazer(uroot3, reg2, no.space=TRUE, type="text")

## 
## ==============================================================
##                                Dependent variable:            
##                     ------------------------------------------
##                                        cinf                   
##                             (1)                   (2)         
## --------------------------------------------------------------
## inf_1                    -0.310***                            
##                           (0.103)                             
## L(cinf, 1)                 0.138                              
##                           (0.126)                             
## L(inf, 1)                                      -0.335***      
##                                                 (0.107)       
## Constant                  1.361**               1.277**       
##                           (0.517)               (0.558)       
## --------------------------------------------------------------
## Observations                 47                   48          
## R2                         0.172                 0.175        
## Adjusted R2                0.134                 0.158        
## Residual Std. Error   2.050 (df = 44)       2.356 (df = 46)   
## F Statistic         4.568** (df = 2; 44) 9.790*** (df = 1; 46)
## ==============================================================
## Note:                              *p<0.1; **p<0.05; ***p<0.01

Example 18.4. Unit Root in the Log of U.S. Real Gross Domestic Product

tsinven <- ts(inven)
gdp_reg <- dynlm(ggdp ~ trend(tsinven) + L(log(gdp), 1) + L(ggdp, 1), data = tsinven)
rho_g1 <- 1 + gdp_reg$coef[3]
rho_g1

## L(log(gdp), 1) 
##      0.7903792

gdp_reg2 <- dynlm(ggdp ~ L(log(gdp), 1) + L(ggdp, 1), data = tsinven)
rho_g2 <- 1 + gdp_reg2$coef[2]
rho_g2

## L(log(gdp), 1) 
##      0.9773125

stargazer(gdp_reg, gdp_reg2, no.space=TRUE, type="text")

## 
## ============================================================
##                               Dependent variable:           
##                     ----------------------------------------
##                                       ggdp                  
##                             (1)                  (2)        
## ------------------------------------------------------------
## trend(tsinven)            0.006**                           
##                           (0.003)                           
## L(log(gdp), 1)            -0.210**             -0.023*      
##                           (0.087)              (0.012)      
## L(ggdp, 1)                 0.264                0.167       
##                           (0.165)              (0.168)      
## Constant                  1.651**              0.215**      
##                           (0.666)              (0.100)      
## ------------------------------------------------------------
## Observations                 35                  35         
## R2                         0.268                0.156       
## Adjusted R2                0.197                0.103       
## Residual Std. Error   0.020 (df = 31)      0.021 (df = 32)  
## F Statistic         3.783** (df = 3; 31) 2.959* (df = 2; 32)
## ============================================================
## Note:                            *p<0.1; **p<0.05; ***p<0.01

Example 18.5. Cointegration between Fertility and Personal Exemption

tsfertil3 <- ts(fertil3)
freg1 <- dynlm(gfr ~ t + pe, data=tsfertil3 )
summary(freg1)

## 
## Time series regression with "ts" data:
## Start = 1, End = 72
## 
## Call:
## dynlm(formula = gfr ~ t + pe, data = tsfertil3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -38.659  -9.934   1.841  11.027  22.882 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 109.93016    3.47526  31.632  < 2e-16 ***
## t            -0.90519    0.10899  -8.305 5.53e-12 ***
## pe            0.18666    0.03463   5.391 9.23e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.2 on 69 degrees of freedom
## Multiple R-squared:  0.5002, Adjusted R-squared:  0.4857 
## F-statistic: 34.53 on 2 and 69 DF,  p-value: 4.064e-11

uf <- resid(freg1)

freg2 <- dynlm(cgfr ~ cpe, data=tsfertil3 ) # Regression in levels
summary(freg2)

## 
## Time series regression with "ts" data:
## Start = 2, End = 72
## 
## Call:
## dynlm(formula = cgfr ~ cpe, data = tsfertil3)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -7.980 -2.552 -0.377  1.866 14.854 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.78478    0.50204  -1.563    0.123
## cpe         -0.04268    0.02837  -1.504    0.137
## 
## Residual standard error: 4.221 on 69 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.03176,    Adjusted R-squared:  0.01773 
## F-statistic: 2.263 on 1 and 69 DF,  p-value: 0.137

Regression in levels with a single lag & time trend, manually

cu <- uf - lag(uf, -1)
freg3 <- dynlm(cu ~ L(u, 1) + L(cu, 1) + t, data=tsfertil3)
summary(freg3)

## 
## Time series regression with "ts" data:
## Start = 3, End = 43
## 
## Call:
## dynlm(formula = cu ~ L(u, 1) + L(cu, 1) + t, data = tsfertil3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -25.139  -2.752  -1.105   1.881  24.343 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.18791    2.49481  -1.278    0.209
## L(u, 1)     -3.18030    8.13220  -0.391    0.698
## L(cu, 1)     0.09839    0.16400   0.600    0.552
## t            0.13609    0.09699   1.403    0.169
## 
## Residual standard error: 7.145 on 37 degrees of freedom
## Multiple R-squared:  0.07913,    Adjusted R-squared:  0.004463 
## F-statistic:  1.06 on 3 and 37 DF,  p-value: 0.3779

First difference regression, with two lags (equation 11.27)

freg4 <- dynlm(cgfr ~ cpe + L(cpe, 1) +  L(cpe, 2), data = tsfertil3)
summary(freg4)

## 
## Time series regression with "ts" data:
## Start = 4, End = 72
## 
## Call:
## dynlm(formula = cgfr ~ cpe + L(cpe, 1) + L(cpe, 2), data = tsfertil3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.8307 -2.1842 -0.1912  1.8442 11.4506 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.96368    0.46776  -2.060  0.04339 *  
## cpe         -0.03620    0.02677  -1.352  0.18101    
## L(cpe, 1)   -0.01397    0.02755  -0.507  0.61385    
## L(cpe, 2)    0.10999    0.02688   4.092  0.00012 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.859 on 65 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.2325, Adjusted R-squared:  0.1971 
## F-statistic: 6.563 on 3 and 65 DF,  p-value: 0.0006054

Example 18.6. Cointegrating Parameter for Interest Rates

tsintqrt <- ts(intqrt)
coreg1 <- dynlm(r6 ~ r3 + cr3 + L(cr3, 1) + L(cr3, 2) + L(cr3, -1) + L(cr3, -2), data = tsintqrt)
summary(coreg1)

## 
## Time series regression with "ts" data:
## Start = 4, End = 122
## 
## Call:
## dynlm(formula = r6 ~ r3 + cr3 + L(cr3, 1) + L(cr3, 2) + L(cr3, 
##     -1) + L(cr3, -2), data = tsintqrt)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.95221 -0.13700 -0.02553  0.11817  0.99889 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.065146   0.056952   1.144  0.25512    
## r3           1.038171   0.008077 128.529  < 2e-16 ***
## cr3         -0.053123   0.019441  -2.733  0.00730 ** 
## L(cr3, 1)   -0.061137   0.019043  -3.210  0.00173 ** 
## L(cr3, 2)   -0.043778   0.018903  -2.316  0.02239 *  
## L(cr3, -1)  -0.003572   0.019122  -0.187  0.85215    
## L(cr3, -2)   0.012366   0.018970   0.652  0.51582    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2455 on 112 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.9942, Adjusted R-squared:  0.9938 
## F-statistic:  3176 on 6 and 112 DF,  p-value: < 2.2e-16

#Test serial correlation
uc <- resid(coreg1)
coreg2 <- dynlm(r6 ~ r3 + cr3 + L(cr3, 1) + L(cr3, 2) + L(cr3, -1) + L(cr3, -2) + L(uc, 1), data = tsintqrt)
summary(coreg2)

## 
## Time series regression with "ts" data:
## Start = 5, End = 122
## 
## Call:
## dynlm(formula = r6 ~ r3 + cr3 + L(cr3, 1) + L(cr3, 2) + L(cr3, 
##     -1) + L(cr3, -2) + L(uc, 1), data = tsintqrt)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.94525 -0.12513 -0.03182  0.11479  1.01592 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.067599   0.057711   1.171  0.24399    
## r3           1.037809   0.008159 127.198  < 2e-16 ***
## cr3         -0.052997   0.019539  -2.712  0.00776 ** 
## L(cr3, 1)   -0.060406   0.019224  -3.142  0.00215 ** 
## L(cr3, 2)   -0.043814   0.018991  -2.307  0.02292 *  
## L(cr3, -1)  -0.003852   0.019213  -0.200  0.84148    
## L(cr3, -2)   0.010436   0.019203   0.543  0.58792    
## L(uc, 1)     0.103944   0.096191   1.081  0.28224    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2464 on 110 degrees of freedom
## Multiple R-squared:  0.9942, Adjusted R-squared:  0.9938 
## F-statistic:  2692 on 7 and 110 DF,  p-value: < 2.2e-16

summary(dynlm(uc ~ L(uc, 1)))

## 
## Time series regression with "ts" data:
## Start = 5, End = 122
## 
## Call:
## dynlm(formula = uc ~ L(uc, 1))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.9384 -0.1248 -0.0324  0.1139  1.0145 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.415e-05  2.209e-02   0.002    0.998
## L(uc, 1)    1.032e-01  9.340e-02   1.105    0.272
## 
## Residual standard error: 0.24 on 116 degrees of freedom
## Multiple R-squared:  0.01041,    Adjusted R-squared:  0.00188 
## F-statistic:  1.22 on 1 and 116 DF,  p-value: 0.2716

# Compare with Simple OLS
summary(lm(r6 ~ r3, data = intqrt))

## 
## Call:
## lm(formula = r6 ~ r3, data = intqrt)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.1379 -0.1583 -0.0251  0.1155  1.1803 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.135374   0.054867   2.467    0.015 *  
## r3          1.025899   0.007709 133.081   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2584 on 122 degrees of freedom
## Multiple R-squared:  0.9932, Adjusted R-squared:  0.9931 
## F-statistic: 1.771e+04 on 1 and 122 DF,  p-value: < 2.2e-16

Example 18.7. Error Correction Model for Holding Yields

tsintqrt <- ts(intqrt)
erreg <- dynlm(chy6 ~ L(chy3, 1) + L(hy6hy3_1, 1), data = tsintqrt)
stargazer(erreg, no.space=TRUE, type="text")

## 
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                                chy6            
## -----------------------------------------------
## L(chy3, 1)                   1.218***          
##                               (0.264)          
## L(hy6hy3_1, 1)               -0.840***         
##                               (0.244)          
## Constant                      0.090**          
##                               (0.043)          
## -----------------------------------------------
## Observations                    122            
## R2                             0.790           
## Adjusted R2                    0.786           
## Residual Std. Error      0.340 (df = 119)      
## F Statistic          223.789*** (df = 2; 119)  
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01

Example 18.8. Forecasting the U.S. Unemployment Rate

tsphillips <- ts(phillips)
reg1 <- lm(unem ~ unem_1, data = tsphillips)
forreg1 <- dynlm(unem ~ unem_1 + inf_1, data = tsphillips)

forcast97 <- forreg1$coef[1] + forreg1$coef[2]*5.4 + forreg1$coef[3]*3 
forcast97 # Forcast of unemp for 1997

## (Intercept) 
##    5.352627

unem_1f <- phillips$unem_1 -  5.4
inf_1f <- phillips$inf_1 -  3
phillips2 <- (cbind (phillips, unem_1f, inf_1f))
reg2 <- lm(unem ~ unem_1f + inf_1f, data = phillips2)
reg2

## 
## Call:
## lm(formula = unem ~ unem_1f + inf_1f, data = phillips2)
## 
## Coefficients:
## (Intercept)      unem_1f       inf_1f  
##      5.3526       0.6495       0.1830

stargazer(reg1, forreg1, reg2, keep.stat=c("n","rsq", "adj.rsq"), no.space=TRUE, type="text")

## 
## ==========================================
##                   Dependent variable:     
##              -----------------------------
##                          unem             
##                 OLS     dynamic     OLS   
##                         linear            
##                 (1)       (2)       (3)   
## ------------------------------------------
## unem_1       0.742***  0.649***           
##               (0.089)   (0.078)           
## inf_1                  0.183***           
##                         (0.039)           
## unem_1f                          0.649*** 
##                                   (0.078) 
## inf_1f                           0.183*** 
##                                   (0.039) 
## Constant     1.490***  1.296***  5.353*** 
##               (0.520)   (0.441)   (0.119) 
## ------------------------------------------
## Observations    55        55        55    
## R2             0.566     0.696     0.696  
## Adjusted R2    0.558     0.685     0.685  
## ==========================================
## Note:          *p<0.1; **p<0.05; ***p<0.01

Example 18.9. Out-of-Sample Comparisons of Unemployment Forecasts

phillips2 <- subset(phillips, phillips$year<1997)
oreg <- lm(unem ~ unem_1, data = phillips2)
RMSE <- function(error) { sqrt(mean(error^2))}
MSE <- function(error) { mean(abs(error))}
RMSE(oreg$residuals)

## [1] 1.026491

MSE(oreg$residuals)

## [1] 0.8129182

summary(abs(resid(oreg)))

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.09675 0.29792 0.72397 0.81292 1.05564 2.82708

oreg2 <- lm(unem ~ unem_1 + inf_1, data = phillips2)
RMSE(oreg2$residuals)

## [1] 0.8549463

MSE(oreg2$residuals)

## [1] 0.6493123

summary(abs(resid(oreg2)))

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.0124  0.2533  0.5500  0.6493  0.8548  2.1730

stargazer(oreg, oreg2, keep.stat=c("n","rsq", "adj.rsq"), no.space=TRUE, type="text")

## 
## =========================================
##                  Dependent variable:     
##              ----------------------------
##                          unem            
##                   (1)            (2)     
## -----------------------------------------
## unem_1          0.732***      0.647***   
##                 (0.097)        (0.084)   
## inf_1                         0.184***   
##                                (0.041)   
## Constant        1.572***       1.304**   
##                 (0.577)        (0.490)   
## -----------------------------------------
## Observations       48            48      
## R2               0.554          0.691    
## Adjusted R2      0.544          0.677    
## =========================================
## Note:         *p<0.1; **p<0.05; ***p<0.01

Example 18.10. Two-Year-Ahead Forecast for the Unemployment Rate

phillips2 <- subset(phillips, phillips$year<1997)
ifreg <- lm(inf ~ inf_1 , data = phillips2)
ifreg2 <- lm(inf ~ inf_1 + unem_1, data = phillips2)

stargazer(ifreg, ifreg2, keep.stat=c("n","rsq", "adj.rsq"), no.space=TRUE, type="text")

## 
## =========================================
##                  Dependent variable:     
##              ----------------------------
##                          inf             
##                   (1)            (2)     
## -----------------------------------------
## inf_1           0.665***      0.659***   
##                 (0.107)        (0.111)   
## unem_1                          0.057    
##                                (0.226)   
## Constant        1.277**         0.974    
##                 (0.558)        (1.320)   
## -----------------------------------------
## Observations       48            48      
## R2               0.457          0.457    
## Adjusted R2      0.445          0.433    
## =========================================
## Note:         *p<0.1; **p<0.05; ***p<0.01

inf96 <- subset(phillips2$inf,phillips2$year==1996)
inf97 <- ifreg$coef[1] + ifreg$coef[2]*inf96
mean(inf97)

## [1] 3.272426

oreg2 <- lm(unem ~ unem_1 + inf_1, data = phillips2)
unem96 <- subset(phillips2$unem,phillips2$year==1996)
unem97 <- oreg2$coef[1] + oreg2$coef[2]*unem96 + oreg2$coef[3]*inf96
unem98 <- (oreg2$coef[1] + oreg2$coef[2]*unem97 + oreg2$coef[3]*inf97)
mean(unem98)

## [1] 5.365136

Introductory Econometrics Using R

Chapter 18. Advanced Time Series Topics

April 19, 2020

Example 18.1. Housing Investment and Residential Price Inflation

Example 18.2. Unit Root Test for Three-Month T-Bill Rates

Example 18.3. Unit Root Test for Annual U.S. Inflation

Example 18.4. Unit Root in the Log of U.S. Real Gross Domestic Product

Example 18.5. Cointegration between Fertility and Personal Exemption

Regression in levels with a single lag & time trend, manually

First difference regression, with two lags (equation 11.27)

Example 18.6. Cointegrating Parameter for Interest Rates

Example 18.7. Error Correction Model for Holding Yields

Example 18.8. Forecasting the U.S. Unemployment Rate

Example 18.9. Out-of-Sample Comparisons of Unemployment Forecasts

Example 18.10. Two-Year-Ahead Forecast for the Unemployment Rate

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