---
title: "Replicating an Empirical Example of International Trade"
output: rmarkdown::html_vignette
bibliography: lit.bib
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{Replicating an Empirical Example of International Trade}
  %\VignetteEncoding{UTF-8}
---


## Introduction

In econometrics, fixed effects models are popular to control for unobserved heterogeneity in data sets with a panel structure. In non-linear models this is usually done by including a dummy variable for each level of a fixed effects category. This approach can quickly become infeasible if the number of levels and/or fixed effects categories increases (either due to memory or time limitations).

One well known example where those limits take place are structural gravity models of trade. In order to demonstrate the capablilities of `feglm()` we replicate parts of @lwyz19 who themselves re-asseses @gr16. Our example is based on a panel data set of 2,973,168 bilateral trade flows. Estimating a theory consistent gravity model with this data set requires to specify a three-way error component with 56,608 levels.

## Data preparation

The data set is available either from [Andrew Rose's website][1] or from [sciencedirect][2]. We use the same variable names as @gr16 such that we are able to compare our summary statistics with the ones provided in their *Stata* log-files.

[1]: https://www.andrewkrose.net/
[2]: https://www.sciencedirect.com/science/article/pii/S0014292116300630#ec0005

```{r, eval = FALSE}
# Import the data set
library(haven)
library(data.table)
cudata <- read_dta("dataaxj1.dta")
setDT(cudata)

# Subsetting relevant variables
var.nms <- c("exp1to2", "custrict11", "ldist", "comlang", "border", "regional",
             "comcol", "curcol", "colony", "comctry", "cuwoemu", "emu", "cuc",
             "cty1", "cty2", "year", "pairid")
cudata <- cudata[, ..var.nms]

# Generate identifiers required for structural gravity
cudata[, pairid := factor(pairid)]
cudata[, exp.time := interaction(cty1, year)]
cudata[, imp.time := interaction(cty2, year)]

# Generate dummies for disaggregated currency unions
cudata[, cuau := as.integer(cuc == "au")]
cudata[, cube := as.integer(cuc == "be")]
cudata[, cuca := as.integer(cuc == "ca")]
cudata[, cucf := as.integer(cuc == "cf")]
cudata[, cucp := as.integer(cuc == "cp")]
cudata[, cudk := as.integer(cuc == "dk")]
cudata[, cuea := as.integer(cuc == "ea")]
cudata[, cuec := as.integer(cuc == "ec")]
cudata[, cuem := as.integer(cuc == "em")]
cudata[, cufr := as.integer(cuc == "fr")]
cudata[, cugb := as.integer(cuc == "gb")]
cudata[, cuin := as.integer(cuc == "in")]
cudata[, cuma := as.integer(cuc == "ma")]
cudata[, cuml := as.integer(cuc == "ml")]
cudata[, cunc := as.integer(cuc == "nc")]
cudata[, cunz := as.integer(cuc == "nz")]
cudata[, cupk := as.integer(cuc == "pk")]
cudata[, cupt := as.integer(cuc == "pt")]
cudata[, cusa := as.integer(cuc == "sa")]
cudata[, cusp := as.integer(cuc == "sp")]
cudata[, cuua := as.integer(cuc == "ua")]
cudata[, cuus := as.integer(cuc == "us")]
cudata[, cuwa := as.integer(cuc == "wa")]
cudata[, cuwoo := custrict11]
cudata[cuc %in% c("em", "au", "cf", "ec", "fr", "gb", "in", "us"), cuwoo := 0L]

# Set missing trade flows to zero
cudata[is.na(exp1to2), exp1to2 := 0.0]
```


## Some replications

After preparing the data, we show how to replicate column 3 of table 2 in @lwyz19. In addition to coefficients and robust standard errors, the authors also report standard errors clustered by exporter, importer, and time.

If we want `feglm()` to report standard errors that are clustered by variables, which are not already part of the model itself, we have to additionally provide them using the third part of the `formula` interface. In this example, we have to additionally add identifiers for exporters (`cty1`), importers (`cty2`), and time (`year`). 

First we report robust standard errors indicated by the option `"sandwich"` in `summary()`.
```{r, eval = FALSE}
mod <- feglm(exp1to2 ~ emu + cuwoo + cuau + cucf + cuec + cufr + cugb + cuin + cuus +
               regional + curcol | exp.time + imp.time + pairid | cty1 + cty2 + year, cudata,
             family = poisson())
summary(mod, "sandwich")
```

```{r, eval = FALSE}
## poisson - log link
## 
## exp1to2 ~ emu + cuwoo + cuau + cucf + cuec + cufr + cugb + cuin + 
##     cuus + regional + curcol | exp.time + imp.time + pairid | 
##     cty1 + cty2 + year
## 
## Estimates:
##            Estimate Std. error z value Pr(> |z|)    
## emu       0.0488950  0.0006057  80.722   < 2e-16 ***
## cuwoo     0.7659882  0.0047822 160.176   < 2e-16 ***
## cuau      0.3844686  0.0562134   6.839  7.95e-12 ***
## cucf     -0.1256085  0.0231247  -5.432  5.58e-08 ***
## cuec     -0.8773179  0.0234656 -37.387   < 2e-16 ***
## cufr      2.0957255  0.0048882 428.730   < 2e-16 ***
## cugb      1.0599574  0.0021330 496.925   < 2e-16 ***
## cuin      0.1697449  0.0068729  24.698   < 2e-16 ***
## cuus      0.0183233  0.0025739   7.119  1.09e-12 ***
## regional  0.1591810  0.0003433 463.662   < 2e-16 ***
## curcol    0.3868821  0.0022466 172.208   < 2e-16 ***
## ---
## Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## 
## residual deviance= 35830779,
## null deviance= 2245707302,
## n= 1610165, l= [11227, 11277, 34104]
## 
## ( 1363003 observation(s) deleted due to perfect classification )
## 
## Number of Fisher Scoring Iterations: 13
```

We observe that roughly 1,4 million observations do not contribute to the identification of the structural parameters and we end up with roughly 1,6 million observations and 57,000 fixed effects.

Replicating the clustered standard errors is straightforward. We simply have to change the option to `"clustered"` and provide `summary` with the requested cluster dimensions.
```{r, eval = FALSE}
summary(mod, "clustered", cluster = ~ cty1 + cty2 + year)
```

```{r, eval = FALSE}
## poisson - log link
## 
## exp1to2 ~ emu + cuwoo + cuau + cucf + cuec + cufr + cugb + cuin + 
##     cuus + regional + curcol | exp.time + imp.time + pairid | 
##     cty1 + cty2 + year
## 
## Estimates:
##          Estimate Std. error z value Pr(> |z|)    
## emu       0.04890    0.09455   0.517   0.60507    
## cuwoo     0.76599    0.24933   3.072   0.00213 ** 
## cuau      0.38447    0.22355   1.720   0.08546 .  
## cucf     -0.12561    0.35221  -0.357   0.72137    
## cuec     -0.87732    0.29493  -2.975   0.00293 ** 
## cufr      2.09573    0.30625   6.843  7.75e-12 ***
## cugb      1.05996    0.23766   4.460  8.19e-06 ***
## cuin      0.16974    0.30090   0.564   0.57267    
## cuus      0.01832    0.05092   0.360   0.71898    
## regional  0.15918    0.07588   2.098   0.03593 *  
## curcol    0.38688    0.15509   2.495   0.01261 *  
## ---
## Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## 
## residual deviance= 35830779,
## null deviance= 2245707302,
## n= 1610165, l= [11227, 11277, 34104]
## 
## ( 1363003 observation(s) deleted due to perfect classification )
## 
## Number of Fisher Scoring Iterations: 13
```


## Testing linear restrictions

Our package is also compatible with `linearHypothesis()` of the `car` package. In the next example we show how to test if all currency union effects except being in the EMU are jointly different from zero using a Wald test based on a clustered covariance matrix.
```{r, eval = FALSE}
library(car)
cus <- c("cuwoo", "cuau", "cucf", "cuec", "cufr", "cugb", "cuin", "cuus")
linearHypothesis(mod, cus, vcov. = vcov(mod, "clustered", cluster = ~ cty1 + cty2 + year))
```


```{r, eval = FALSE}
## Linear hypothesis test
## 
## Hypothesis:
## cuwoo = 0
## cuau = 0
## cucf = 0
## cuec = 0
## cufr = 0
## cugb = 0
## cuin = 0
## cuus = 0
## 
## Model 1: restricted model
## Model 2: exp1to2 ~ emu + cuwoo + cuau + cucf + cuec + cufr + cugb + cuin + 
##     cuus + regional + curcol | exp.time + imp.time + pairid | 
##     cty1 + cty2 + year
## 
## Note: Coefficient covariance matrix supplied.
## 
##   Df  Chisq Pr(>Chisq)    
## 1                         
## 2  8 96.771  < 2.2e-16 ***
## ---
## Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
```


## References