Last updated: 2020-12-29
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | 62aa447 | KaranSShakya | 2020-12-28 | 8731 qt3 |
| html | 62aa447 | KaranSShakya | 2020-12-28 | 8731 qt3 |
| Rmd | 97b8329 | KaranSShakya | 2020-12-28 | 8731 qt 2 |
| html | 97b8329 | KaranSShakya | 2020-12-28 | 8731 qt 2 |
| Rmd | bd2489c | KaranSShakya | 2020-12-27 | 8731 q1 half solved |
| html | bd2489c | KaranSShakya | 2020-12-27 | 8731 q1 half solved |
| Rmd | b07f4c3 | KaranSShakya | 2020-12-27 | econometrics problems page + data |
| html | b07f4c3 | KaranSShakya | 2020-12-27 | econometrics problems page + data |
Dataset is:
# A tibble: 2 x 5
Obs Year Quarter Income Consumption
<dbl> <dbl> <dbl> <dbl> <dbl>
1 1 1947 1 59505 57168
2 2 1947 2 59717. 55464\(c_t\) is log of consumption, \(y_t\) is log of income. Their first differences are \(\Delta c_t = c_t - c_{t-1}\) and \(\Delta y_t = y_t - y_{t-1}\).
The model for the period 1953:1 to 1996:4, \[ \Delta c_t = \beta_1 + \beta_2 \Delta y_t + \beta_3 \Delta y_{t-1} + \beta_4 \Delta y_{t-2} + \beta_5 \Delta y_{t-3} + \beta_6 \Delta y_{t-4} \] ## Estimation Let \(\gamma = \sum_{l=2}^{6} \beta_l\). Calculate \(\hat{\gamma}\) and its SE.
consumption1 <- consumption %>%
mutate(ct = log(Consumption), yt=log(Income),
del_ct = ct-lag(ct),
del_yt = yt-lag(yt),
yt1 = lag(yt, 1),
yt2 = lag(yt, 2),
yt3 = lag(yt, 3),
yt4 = lag(yt, 4),
del_yt1 = yt1-lag(yt1),
del_yt2 = yt2-lag(yt2),
del_yt3 = yt3-lag(yt3),
del_yt4 = yt4-lag(yt4),
ct1 = lag(ct, 1)) #adding new variables
cons1 <- consumption1 %>%
filter(Year >= 1953 & Year <= 1996)
cons1.model <- lm(del_ct ~ del_yt+del_yt1+del_yt2+del_yt3+del_yt4, data=cons1)
summ(cons1.model, digits = 4)MODEL INFO:
Observations: 176
Dependent Variable: del_ct
Type: OLS linear regression
MODEL FIT:
F(5,170) = 10.6489, p = 0.0000
R² = 0.2385
Adj. R² = 0.2161
Standard errors: OLS
-------------------------------------------------------
Est. S.E. t val. p
----------------- --------- -------- --------- --------
(Intercept) 0.0051 0.0012 4.3589 0.0000
del_yt 0.3058 0.0545 5.6117 0.0000
del_yt1 0.1242 0.0544 2.2813 0.0238
del_yt2 0.0425 0.0551 0.7724 0.4410
del_yt3 0.1239 0.0545 2.2730 0.0243
del_yt4 -0.1233 0.0539 -2.2883 0.0234
-------------------------------------------------------gamma_h <- cons1.model$coefficients[2]+cons1.model$coefficients[3]+
cons1.model$coefficients[4]+cons1.model$coefficients[5]+
cons1.model$coefficients[6]
kable(gamma_h, row.names = F, align = "l")| x |
|---|
| 0.4731631 |
The dataset is as follows:
# A tibble: 5 x 4
Obs Dep X_2 X_3
<dbl> <dbl> <dbl> <dbl>
1 1 2.75 0.0121 0.0365
2 2 8.23 1.69 3.74
3 3 6.70 0.404 2.31
4 4 7.28 0.606 1.62
5 5 7.64 2.11 6.95 Classical linear regression model, \[ \begin{align} y = \beta_1 + \beta_2 x_2 + \beta_3 x_3 + u, && u \sim N (0, \sigma^2 I) \end{align} \]
model.class <- lm(Dep~X_2+X_3, data = classical)
summ(model.class, digits = 4)MODEL INFO:
Observations: 50
Dependent Variable: Dep
Type: OLS linear regression
MODEL FIT:
F(2,47) = 68.1609, p = 0.0000
R² = 0.7436
Adj. R² = 0.7327
Standard errors: OLS
------------------------------------------------------
Est. S.E. t val. p
----------------- -------- -------- --------- --------
(Intercept) 5.0949 0.1976 25.7815 0.0000
X_2 1.1349 0.1408 8.0583 0.0000
X_3 0.2221 0.0945 2.3487 0.0231
------------------------------------------------------test \(\beta_3 = 0\)
linearHypothesis(model.class, "X_3 = 0")Linear hypothesis test
Hypothesis:
X_3 = 0
Model 1: restricted model
Model 2: Dep ~ X_2 + X_3
Res.Df RSS Df Sum of Sq F Pr(>F)
1 48 44.114
2 47 39.481 1 4.6339 5.5165 0.02309 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Or we could have simply looked at the t-value provided at the summ table. The p-values are the same for both.
Warning in remove(classical, model.class, X_3): object 'X_3' not foundDummy variable dataset, where Group 1, Group 2, and Group 3 are dummy variables.
# A tibble: 3 x 5
Obs Group1 Group2 Group3 Earnings
<dbl> <dbl> <dbl> <dbl> <dbl>
1 1 0 0 1 570.
2 2 0 0 1 896.
3 3 0 0 1 1111 Regress y on all 3 dummy variables. Important to remove the intercept term. And construct a 0.95 CI of Group3:
model.dum <- lm(Earnings~-1+Group1+Group2+Group3, data=earnings)
summ(model.dum, confint = T, ci.width = 0.95, digits = 2)MODEL INFO:
Observations: 4266
Dependent Variable: Earnings
Type: OLS linear regression
MODEL FIT:
F(3,4263) = 3842.49, p = 0.00
R² = 0.73
Adj. R² = 0.73
Standard errors: OLS
-------------------------------------------------------------
Est. 2.5% 97.5% t val. p
------------ ---------- ---------- ---------- -------- ------
Group1 22880.48 21964.03 23796.92 48.95 0.00
Group2 25080.17 24335.14 25825.21 66.00 0.00
Group3 27973.63 27180.06 28767.21 69.11 0.00
-------------------------------------------------------------Regress y on Group3 only:
model.dum3 <- lm(Earnings~-1+Group3, data=earnings)
summ(model.dum3, digits = 2, confint = T, ci.width = 0.95, robust = "HC1")MODEL INFO:
Observations: 4266
Dependent Variable: Earnings
Type: OLS linear regression
MODEL FIT:
F(1,4265) = 1849.35, p = 0.00
R² = 0.30
Adj. R² = 0.30
Standard errors: Robust, type = HC1
-------------------------------------------------------------
Est. 2.5% 97.5% t val. p
------------ ---------- ---------- ---------- -------- ------
Group3 27973.63 27132.54 28814.73 65.20 0.00
-------------------------------------------------------------We start off with the regression:
model.gas <- lm(Gas_pop~Year+GasP+Income+Pnc+Puc+Ppt+Pd+Pn+Ps,
data=gasoline)
summ(model.gas, digits = 10)MODEL INFO:
Observations: 52
Dependent Variable: Gas_pop
Type: OLS linear regression
MODEL FIT:
F(9,42) = 9827.1336396413, p = 0.0000000000
R² = 0.9995253497
Adj. R² = 0.9994236390
Standard errors: OLS
-------------------------------------------------------------------------------
Est. S.E. t val. p
----------------- --------------- -------------- --------------- --------------
(Intercept) 0.0038179534 0.0011927155 3.2010594676 0.0026091196
Year -0.0000020401 0.0000006219 -3.2803740791 0.0020897345
GasP 0.0000050346 0.0000001744 28.8615597254 0.0000000000
Income 0.0000000147 0.0000000023 6.4728930994 0.0000000829
Pnc 0.0000017334 0.0000005632 3.0777460893 0.0036654373
Puc 0.0000006405 0.0000002136 2.9988805264 0.0045399261
Ppt 0.0000000137 0.0000002121 0.0644989404 0.9488789686
Pd -0.0000030835 0.0000005210 -5.9183360943 0.0000005206
Pn 0.0000009728 0.0000005524 1.7609219249 0.0855314029
Ps -0.0000005597 0.0000003506 -1.5961901256 0.1179443038
-------------------------------------------------------------------------------The log transformation for elasticity:
model.logg <- lm(log(Gas_pop)~Year+log(GasP)+
log(Income)+log(Pnc)+log(Puc)+log(Ppt)+log(Pd)+
log(Pn)+log(Ps), data=gasoline)
summ(model.logg, digits = 5)MODEL INFO:
Observations: 52
Dependent Variable: log(Gas_pop)
Type: OLS linear regression
MODEL FIT:
F(9,42) = 4950.26087, p = 0.00000
R² = 0.99906
Adj. R² = 0.99886
Standard errors: OLS
-------------------------------------------------------------
Est. S.E. t val. p
----------------- ----------- ---------- ---------- ---------
(Intercept) -95.22401 13.11888 -7.25855 0.00000
Year 0.03797 0.00751 5.05370 0.00001
log(GasP) 1.06052 0.05401 19.63552 0.00000
log(Income) 0.99299 0.25038 3.96600 0.00028
log(Pnc) -0.15472 0.26696 -0.57954 0.56532
log(Puc) -0.48909 0.08520 -5.74053 0.00000
log(Ppt) 0.01927 0.13645 0.14122 0.88837
log(Pd) 1.73206 0.25989 6.66468 0.00000
log(Pn) -0.72954 0.26507 -2.75227 0.00870
log(Ps) -0.86798 0.35291 -2.45949 0.01811
-------------------------------------------------------------The dataset is:
# A tibble: 3 x 14
Id Year Cost Q Pl Sl Pk Sk Pf Sf C_Pf ln_q
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 1 1970 0.213 8 6869. 0.329 64.9 0.420 18 0.251 0.0118 2.16
2 4 1970 3.04 869 8373. 0.103 68.2 0.291 21.1 0.606 0.144 22.9
3 5 1970 9.41 1412 7961. 0.0891 40.7 0.157 41.5 0.754 0.226 26.3
# … with 2 more variables: Pk_Pf <dbl>, Pl_Pf <dbl>The model is:
model.c <- lm(log(C_Pf) ~ log(Q)+ln_q+log(Pk_Pf)+log(Pl_Pf), data=cost)
summ(model.c, digits = 6)MODEL INFO:
Observations: 158
Dependent Variable: log(C_Pf)
Type: OLS linear regression
MODEL FIT:
F(4,153) = 4879.590243, p = 0.000000
R² = 0.992222
Adj. R² = 0.992019
Standard errors: OLS
----------------------------------------------------------------
Est. S.E. t val. p
----------------- ----------- ---------- ------------ ----------
(Intercept) -6.818163 0.252439 -27.009131 0.000000
log(Q) 0.402745 0.031483 12.792425 0.000000
ln_q 0.060895 0.004325 14.078838 0.000000
log(Pk_Pf) 0.162034 0.040406 4.010187 0.000095
log(Pl_Pf) 0.152445 0.046597 3.271532 0.001322
----------------------------------------------------------------Estimate the covariance matrix:
vcov(model.c) (Intercept) log(Q) ln_q log(Pk_Pf)
(Intercept) 0.0637255474 -0.0023838181 3.104204e-04 3.994585e-03
log(Q) -0.0023838181 0.0009911868 -1.335824e-04 1.002255e-04
ln_q 0.0003104204 -0.0001335824 1.870819e-05 -1.493338e-05
log(Pk_Pf) 0.0039945854 0.0001002255 -1.493338e-05 1.632609e-03
log(Pl_Pf) -0.0104712922 -0.0001995679 2.453652e-05 -1.019813e-03
log(Pl_Pf)
(Intercept) -1.047129e-02
log(Q) -1.995679e-04
ln_q 2.453652e-05
log(Pk_Pf) -1.019813e-03
log(Pl_Pf) 2.171313e-03
sessionInfo()R version 4.0.0 (2020-04-24)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] AER_1.2-9 survival_3.1-12 sandwich_3.0-0 lmtest_0.9-38
[5] zoo_1.8-8 car_3.0-10 carData_3.0-4 gridExtra_2.3
[9] GGally_2.0.0 jtools_2.1.1 stargazer_5.2.2 broom_0.5.6
[13] readxl_1.3.1 forcats_0.5.0 stringr_1.4.0 dplyr_0.8.5
[17] purrr_0.3.4 readr_1.3.1 tidyr_1.0.3 tibble_3.0.1
[21] ggplot2_3.3.0 tidyverse_1.3.0 knitr_1.28 workflowr_1.6.2
loaded via a namespace (and not attached):
[1] nlme_3.1-147 fs_1.4.1 lubridate_1.7.8 RColorBrewer_1.1-2
[5] httr_1.4.1 rprojroot_1.3-2 tools_4.0.0 backports_1.1.6
[9] utf8_1.1.4 R6_2.4.1 DBI_1.1.0 colorspace_1.4-1
[13] withr_2.2.0 tidyselect_1.1.0 curl_4.3 compiler_4.0.0
[17] git2r_0.27.1 cli_2.0.2 rvest_0.3.5 xml2_1.3.2
[21] scales_1.1.1 digest_0.6.25 foreign_0.8-78 rmarkdown_2.6
[25] rio_0.5.16 pkgconfig_2.0.3 htmltools_0.5.0 highr_0.8
[29] dbplyr_1.4.3 rlang_0.4.6 rstudioapi_0.11 generics_0.0.2
[33] jsonlite_1.6.1 zip_2.0.4 magrittr_1.5 Formula_1.2-3
[37] Matrix_1.2-18 Rcpp_1.0.4.6 munsell_0.5.0 fansi_0.4.1
[41] abind_1.4-5 lifecycle_0.2.0 stringi_1.4.6 whisker_0.4
[45] yaml_2.2.1 plyr_1.8.6 grid_4.0.0 promises_1.1.0
[49] crayon_1.3.4 lattice_0.20-41 haven_2.2.0 splines_4.0.0
[53] pander_0.6.3 hms_0.5.3 pillar_1.4.4 reprex_0.3.0
[57] glue_1.4.1 evaluate_0.14 data.table_1.12.8 modelr_0.1.7
[61] vctrs_0.3.0 httpuv_1.5.2 cellranger_1.1.0 gtable_0.3.0
[65] reshape_0.8.8 assertthat_0.2.1 xfun_0.19 openxlsx_4.1.5
[69] later_1.0.0 ellipsis_0.3.0