Chapter 10: Basic Regression Analysis with Time Series Data

Franz X. Mohr, Created: October 7, 2018, Last update: October 7, 2018

library(wooldridge)

Example 10.1

data("phillips")

lm.10.1 <- lm(inf ~ unem, data = phillips)
summary(lm.10.1)

## 
## Call:
## lm(formula = inf ~ unem, data = phillips)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.2176 -1.7812 -0.6659  1.1473  8.8795 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   1.0536     1.5480   0.681   0.4990  
## unem          0.5024     0.2656   1.892   0.0639 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.972 on 54 degrees of freedom
## Multiple R-squared:  0.06215,    Adjusted R-squared:  0.04479 
## F-statistic: 3.579 on 1 and 54 DF,  p-value: 0.06389

Example 10.2

data("intdef")

lm.10.2 <- lm(i3 ~ inf + def, data = intdef)
summary(lm.10.2)

## 
## Call:
## lm(formula = i3 ~ inf + def, data = intdef)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9948 -1.1694  0.1959  0.9602  4.7224 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.73327    0.43197   4.012  0.00019 ***
## inf          0.60587    0.08213   7.376 1.12e-09 ***
## def          0.51306    0.11838   4.334 6.57e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.843 on 53 degrees of freedom
## Multiple R-squared:  0.6021, Adjusted R-squared:  0.5871 
## F-statistic: 40.09 on 2 and 53 DF,  p-value: 2.483e-11

Example 10.3

data("prminwge")

lm.10.3 <- lm(lprepop ~ lmincov + lusgnp, data = prminwge)
summary(lm.10.3)

## 
## Call:
## lm(formula = lprepop ~ lmincov + lusgnp, data = prminwge)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.117133 -0.036998 -0.005943  0.028182  0.113938 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -1.05442    0.76541  -1.378   0.1771  
## lmincov     -0.15444    0.06490  -2.380   0.0229 *
## lusgnp      -0.01219    0.08851  -0.138   0.8913  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0557 on 35 degrees of freedom
## Multiple R-squared:  0.6605, Adjusted R-squared:  0.6411 
## F-statistic: 34.04 on 2 and 35 DF,  p-value: 6.17e-09

Example 10.4

data("fertil3")

lm.10.4.1 <- lm(gfr ~ pe + ww2 + pill, data = fertil3)
summary(lm.10.4.1)

## 
## Call:
## lm(formula = gfr ~ pe + ww2 + pill, data = fertil3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -27.0187  -9.6195   0.3393   9.4746  28.0730 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  98.68176    3.20813  30.760  < 2e-16 ***
## pe            0.08254    0.02965   2.784  0.00694 ** 
## ww2         -24.23840    7.45825  -3.250  0.00180 ** 
## pill        -31.59403    4.08107  -7.742 6.46e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.69 on 68 degrees of freedom
## Multiple R-squared:  0.4734, Adjusted R-squared:  0.4502 
## F-statistic: 20.38 on 3 and 68 DF,  p-value: 1.575e-09

lm.10.4.2 <- lm(gfr ~ pe + pe_1 + pe_2 + ww2 + pill, data = fertil3)
summary(lm.10.4.2)

## 
## Call:
## lm(formula = gfr ~ pe + pe_1 + pe_2 + ww2 + pill, data = fertil3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -24.6461  -9.5409  -0.0312   8.3378  29.1295 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  95.87050    3.28196  29.211  < 2e-16 ***
## pe            0.07267    0.12553   0.579   0.5647    
## pe_1         -0.00578    0.15566  -0.037   0.9705    
## pe_2          0.03383    0.12626   0.268   0.7896    
## ww2         -22.12650   10.73197  -2.062   0.0433 *  
## pill        -31.30499    3.98156  -7.862 5.63e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.27 on 64 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.4986, Adjusted R-squared:  0.4594 
## F-statistic: 12.73 on 5 and 64 DF,  p-value: 1.353e-08

# Joint significance of pe values
lm.10.4.2res <- lm(gfr ~ ww2 + pill, data = fertil3, subset = (is.na(pe_2) == FALSE))
anova(lm.10.4.2, lm.10.4.2res)

## Analysis of Variance Table
## 
## Model 1: gfr ~ pe + pe_1 + pe_2 + ww2 + pill
## Model 2: gfr ~ ww2 + pill
##   Res.Df   RSS Df Sum of Sq     F  Pr(>F)  
## 1     64 13033                             
## 2     67 15460 -3   -2427.1 3.973 0.01165 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Joint significance of lagged pe values
lm.10.4.2res2 <- lm(gfr ~ pe + ww2 + pill, data = fertil3, subset = (is.na(pe_2) == FALSE))
anova(lm.10.4.2, lm.10.4.2res2)

## Analysis of Variance Table
## 
## Model 1: gfr ~ pe + pe_1 + pe_2 + ww2 + pill
## Model 2: gfr ~ pe + ww2 + pill
##   Res.Df   RSS Df Sum of Sq      F Pr(>F)
## 1     64 13033                           
## 2     66 13054 -2   -21.761 0.0534  0.948

# Standard error of the long run propensity
summary(lm(gfr ~ pe + I(pe_1 - pe) + I(pe_2 - pe_1) + ww2 + pill, data = fertil3))

## 
## Call:
## lm(formula = gfr ~ pe + I(pe_1 - pe) + I(pe_2 - pe_1) + ww2 + 
##     pill, data = fertil3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -24.6461  -9.5409  -0.0312   8.3378  29.1295 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     95.87050    3.28196  29.211  < 2e-16 ***
## pe               0.10072    0.02980   3.380  0.00124 ** 
## I(pe_1 - pe)     0.02805    0.11890   0.236  0.81427    
## I(pe_2 - pe_1)   0.03383    0.12626   0.268  0.78962    
## ww2            -22.12650   10.73197  -2.062  0.04330 *  
## pill           -31.30499    3.98156  -7.862 5.63e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.27 on 64 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.4986, Adjusted R-squared:  0.4594 
## F-statistic: 12.73 on 5 and 64 DF,  p-value: 1.353e-08

Example 10.5

data("barium")

lm.10.5 <- lm(lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6, data = barium)
summary(lm.10.5)

## 
## Call:
## lm(formula = lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + 
##     afdec6, data = barium)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.03356 -0.39080  0.03048  0.40248  1.51719 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -17.80300   21.04537  -0.846   0.3992    
## lchempi       3.11719    0.47920   6.505 1.72e-09 ***
## lgas          0.19635    0.90662   0.217   0.8289    
## lrtwex        0.98302    0.40015   2.457   0.0154 *  
## befile6       0.05957    0.26097   0.228   0.8198    
## affile6      -0.03241    0.26430  -0.123   0.9026    
## afdec6       -0.56524    0.28584  -1.978   0.0502 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5974 on 124 degrees of freedom
## Multiple R-squared:  0.3049, Adjusted R-squared:  0.2712 
## F-statistic: 9.064 on 6 and 124 DF,  p-value: 3.255e-08

Example 10.6

data("fair")

lm.10.6 <- lm(demvote ~ partyWH + incum + I(partyWH*gnews) + I(partyWH*inf), data = fair)
summary(lm.10.6)

## 
## Call:
## lm(formula = demvote ~ partyWH + incum + I(partyWH * gnews) + 
##     I(partyWH * inf), data = fair)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.059843 -0.035554 -0.001095  0.015049  0.104859 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         0.484219   0.011472  42.210   <2e-16 ***
## partyWH            -0.032380   0.037550  -0.862   0.4013    
## incum               0.056369   0.023017   2.449   0.0262 *  
## I(partyWH * gnews)  0.009666   0.003807   2.539   0.0219 *  
## I(partyWH * inf)   -0.008338   0.003121  -2.672   0.0167 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.049 on 16 degrees of freedom
## Multiple R-squared:  0.6588, Adjusted R-squared:  0.5735 
## F-statistic: 7.725 on 4 and 16 DF,  p-value: 0.001151

Slightly different results since the book uses 20 observations to 1992. But the results are the same when we drop the last observation.

lm.10.6 <- lm(demvote ~ partyWH + incum + I(partyWH * gnews) + I(partyWH * inf), data = fair, subset = (year != 1996))
summary(lm.10.6)

## 
## Call:
## lm(formula = demvote ~ partyWH + incum + I(partyWH * gnews) + 
##     I(partyWH * inf), data = fair, subset = (year != 1996))
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.059643 -0.036072 -0.005397  0.018386  0.102723 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         0.481062   0.012263  39.228   <2e-16 ***
## partyWH            -0.043475   0.040459  -1.075   0.2996    
## incum               0.054390   0.023417   2.323   0.0347 *  
## I(partyWH * gnews)  0.010847   0.004127   2.628   0.0190 *  
## I(partyWH * inf)   -0.007702   0.003257  -2.365   0.0319 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.04957 on 15 degrees of freedom
## Multiple R-squared:  0.6629, Adjusted R-squared:  0.573 
## F-statistic: 7.374 on 4 and 15 DF,  p-value: 0.001716

Example 10.7

data("hseinv")

lm.10.7.1 <- lm(linvpc ~ lprice, data = hseinv)
summary(lm.10.7.1)

## 
## Call:
## lm(formula = linvpc ~ lprice, data = hseinv)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.45991 -0.08694 -0.01264  0.08651  0.34672 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.55023    0.04303 -12.788 1.03e-15 ***
## lprice       1.24094    0.38242   3.245  0.00238 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1554 on 40 degrees of freedom
## Multiple R-squared:  0.2084, Adjusted R-squared:  0.1886 
## F-statistic: 10.53 on 1 and 40 DF,  p-value: 0.002376

Add a time trend

lm.10.7.2 <- lm(linvpc ~ lprice + t, data = hseinv)
summary(lm.10.7.2)

## 
## Call:
## lm(formula = linvpc ~ lprice + t, data = hseinv)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.45092 -0.08583 -0.01734  0.08517  0.26024 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.913060   0.135613  -6.733    5e-08 ***
## lprice      -0.380961   0.678835  -0.561  0.57787    
## t            0.009829   0.003512   2.798  0.00794 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1436 on 39 degrees of freedom
## Multiple R-squared:  0.3408, Adjusted R-squared:  0.307 
## F-statistic: 10.08 on 2 and 39 DF,  p-value: 0.000296

Example 10.8

data("fertil3")

lm.10.8.1 <- lm(gfr ~ pe + ww2 + pill + t, data = fertil3)
summary(lm.10.8.1)

## 
## Call:
## lm(formula = gfr ~ pe + ww2 + pill + t, data = fertil3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -24.6418  -8.4574   0.2445   9.3819  17.2837 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 111.76943    3.35777  33.287  < 2e-16 ***
## pe            0.27888    0.04002   6.968 1.72e-09 ***
## ww2         -35.59228    6.29738  -5.652 3.53e-07 ***
## pill          0.99745    6.26163   0.159    0.874    
## t            -1.14987    0.18790  -6.119 5.49e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.85 on 67 degrees of freedom
## Multiple R-squared:  0.6622, Adjusted R-squared:  0.642 
## F-statistic: 32.84 on 4 and 67 DF,  p-value: 3.756e-15

lm.10.8.2 <- lm(gfr ~ pe + ww2 + pill + t + tsq, data = fertil3)
summary(lm.10.8.2)

## 
## Call:
## lm(formula = gfr ~ pe + ww2 + pill + t + tsq, data = fertil3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -25.9791  -6.9775  -0.2713   7.7975  19.9861 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 124.091935   4.360738  28.457  < 2e-16 ***
## pe            0.347813   0.040260   8.639 1.91e-12 ***
## ww2         -35.880277   5.707921  -6.286 2.95e-08 ***
## pill        -10.119723   6.336094  -1.597 0.115008    
## t            -2.531426   0.389386  -6.501 1.24e-08 ***
## tsq           0.019613   0.004971   3.945 0.000196 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.74 on 66 degrees of freedom
## Multiple R-squared:  0.7267, Adjusted R-squared:  0.706 
## F-statistic: 35.09 on 5 and 66 DF,  p-value: < 2.2e-16

Example 10.9

data("prminwge")

lm.10.9 <- lm(lprepop ~ lmincov + lusgnp + t, data = prminwge)
summary(lm.10.9)

## 
## Call:
## lm(formula = lprepop ~ lmincov + lusgnp + t, data = prminwge)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.049493 -0.024425 -0.009596  0.017055  0.086354 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -8.696298   1.295764  -6.711 1.04e-07 ***
## lmincov     -0.168695   0.044246  -3.813 0.000552 ***
## lusgnp       1.057351   0.176637   5.986 8.98e-07 ***
## t           -0.032354   0.005023  -6.442 2.31e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03793 on 34 degrees of freedom
## Multiple R-squared:  0.8471, Adjusted R-squared:  0.8336 
## F-statistic: 62.78 on 3 and 34 DF,  p-value: 6.007e-14

# Obtain results by detrending
lprepop.detrend <- lm(lprepop ~ t, data = prminwge)$resid
lmincov.detrend <- lm(lmincov ~ t, data = prminwge)$resid
lusgnp.detrend <- lm(lusgnp ~ t, data = prminwge)$resid

summary(lm(lprepop.detrend ~ lmincov.detrend + lusgnp.detrend - 1))

## 
## Call:
## lm(formula = lprepop.detrend ~ lmincov.detrend + lusgnp.detrend - 
##     1)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.049493 -0.024425 -0.009596  0.017055  0.086354 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## lmincov.detrend  -0.1687     0.0430  -3.923 0.000377 ***
## lusgnp.detrend    1.0574     0.1717   6.160 4.26e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03686 on 36 degrees of freedom
## Multiple R-squared:  0.5327, Adjusted R-squared:  0.5067 
## F-statistic: 20.52 on 2 and 36 DF,  p-value: 1.129e-06

Example 10.10

data("hseinv")

dtr.linvpc <- lm(linvpc ~ t, data = hseinv)$resid
summary(lm(dtr.linvpc ~ lprice + t, data = hseinv))

## 
## Call:
## lm(formula = dtr.linvpc ~ lprice + t, data = hseinv)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.45092 -0.08583 -0.01734  0.08517  0.26024 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.071768   0.135613  -0.529    0.600
## lprice      -0.380961   0.678835  -0.561    0.578
## t            0.001683   0.003512   0.479    0.635
## 
## Residual standard error: 0.1436 on 39 degrees of freedom
## Multiple R-squared:  0.008011,   Adjusted R-squared:  -0.04286 
## F-statistic: 0.1575 on 2 and 39 DF,  p-value: 0.8548

Example 10.11

data("barium")

lm.10.11 <- lm(lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6 +
 feb + mar + apr + may + jun + jul + aug + sep + oct + nov + dec, data = barium)
lm.10.11res <- lm(lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6, data = barium)

anova(lm.10.11, lm.10.11res)

## Analysis of Variance Table
## 
## Model 1: lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6 + 
##     feb + mar + apr + may + jun + jul + aug + sep + oct + nov + 
##     dec
## Model 2: lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6
##   Res.Df    RSS  Df Sum of Sq      F Pr(>F)
## 1    113 40.844                            
## 2    124 44.247 -11   -3.4032 0.8559 0.5852