Intermediate Econometrics

• Let us consider the regression \[y_i = \beta_1 + \beta_2 x_i + \varepsilon_i \]
• Let \(z_i\) an instrumental variable, that is correlated with \(x_i\) but it is uncorrelated with \(\varepsilon_i\)
• We consider the exactly/just identified case (the number of instruments, one in this case, is equal to the number of regressors, one in this case)

• Stage 1: Regress \(X\) on \(Z\) via OLS \[x_i = \pi_1 + \pi_2 z_i + \varepsilon^*_i\] and calculate the predicted values \[\hat x_i = \hat \pi_1 + \hat \pi_2 z_i\]
• Stage 2: In the regression of interest, substitute the observed \(x_i\) with the predicted \(\hat x_i\), \[y_i = \beta_1 + \beta_2 \hat x_i + \varepsilon_i \]

\[Y = X\beta + \varepsilon\]

• Let’s denote with \(Z\) the model matrix of the instruments and with \(X\) the model matrix of the regressors

• The estimator \(\hat \beta^{TSLS}\) is \[\hat \beta^{TSLS} = (X^\top P_Z X)^{-1}X^\top P_Z Y\] where \(P_Z = Z (Z^\top Z)^{-1}Z^\top\)

• It is clear the reason of substituting \(\hat X\) onto \(X\). Indeed, one of the results of the OLS is that \(\hat X = P_Z X\) and so \[\hat \beta = (\hat X \hat X^\top)^{-1}\hat X^\top Y\]

1. Consistent (\(\hat \beta \xrightarrow{p} \beta\))
2. The estimate of the variance of the estimator is, assuming spherical disturbances, equal to \[{\mathbb{\hat V}}(\hat \beta^{TSLS}) = \hat \sigma^2 (X^\top P_Z X)^{-1}\] where \[\hat \sigma^2 = \frac{1}{n-{\rm dim}(\beta)} \sum_{i=1}^{n} e^2_i\] with \(e_i = y_i -X\hat \beta^{TSLS}\)

• From \({\mathbb{\hat V}}(\hat \beta^{TSLS})\) we can obtain the estimate of the standard error of the \(\beta\)’s

• This is an example aiming to investigate the effect of segregation on urban poverty and inequality (it is extracted from Ananat, 2011)

• As outcome variable, it is considered the poverty rate for the black populations of 121 American cities

• The level of segregation is measured by a dissimilarity index. This index vary from 0 (no segregation) to 1 (perfect segregation)

• In that work, the author suggest that the way cities were subdivided by railroads into a large number of neighborhoods can be used as an instrument.

• Moreover, the tracks were mostly built during the nineteenth century, prior to the great migration (between 1915 to 1950) of African Americans from the south.

• So, it is used an index that is 0 if the city is completely undivided and tends to 1 if the number of neighborhoods tends to infinity.

• The dataset is named tracks_side, and it is in .csv format

• From the summary we can see:

1. the poverty rate (povb) ranges from 0.09 to 0.50 with a median value of 0.26;
2. the segregation index (segregation) ranges from 0.33 to 0.87, with a median value of 0.57;
3. the rail tracks index (raildiv) ranges from 0.24 to 0.99 in the sample, with a median value of 0.74;

• We can notice a strong correlation between segregation and raildiv and povb, while a low correlation between raildiv and povb

• We’ll focus on the effect of segregation on the poverty rate of black people: let’s fit the model \[povb_i = \beta_1 + \beta_2 segregation_i + \varepsilon_i\]
• The coefficient of segregation is positive and significant. It indicates that a 1 point increase of the segregation index raises the poverty rate of black people by about 0.18 point

• The correlation between segregation and bad economic outcome for black people is well established but, according to the author, the OLS estimator cannot easily be considered as a measure of the causal relationship of segregation on income, as there are some other variables that both influence segregation and outcome for black people

• As an example, the situation of Detroit is described, which is a highly segregated city with poor economic outcomes, but other characteristics of the city (political corruption, legacy of a manufacturing economy) can be the cause of these two phenomena

• Therefore, the OLS estimator is suspected to be biased and inconsistent because of the omitted variable bias.

• The instrumental variable estimator can be used in this context, but it requires the use of a good instrumental variable, i.e., a variable which is correlated with the endogenous covariate (segregation), but not directly with the response (the poverty rate).

• Regression of segregation on raildiv results in a coefficient of raildiv positive and highly significant

• It is important to check that the correlation between the covariate and the instrument is strong enough to get a precise IV estimator. This can be performed by computing

1. the coefficient of correlation
2. the \(R^2\) or the \(F\) statistic of the first stage regression

• The IV estimator can be obtained by regressing the response on the fitted values of the first stage regression
• The IV estimator is larger than the OLS estimator (0.23 vs. 0.18).
• An increase of 1 point of segregation would increase povb of 0.23 points

• The IV estimate can be obtained by dividing the OLS coefficients of the regressions of y on z and of x on z.
• A 1 point increase of raildiv is associated with a 0.4 point increase of the discrimination index and with a 0.09 point of the poverty rate. Therefore, the 0.4 point increase of segregation increases povb by 0.09, which means that an increase of 1 point of segregation would increase povb by 0.09/0.4 = 0.23

• The IV regression can be obtained easily using the ivreg() function (in the homonimous package)

1. \(\hat \beta^{TSLS} = (X^\top P_Z X)^{-1}X^\top P_Z Y\), where \(P_Z = Z (Z^\top Z)^{-1}Z^\top\)
   
2. \({\mathbb{\hat V}}(\hat \beta) = \hat \sigma^2 (X^\top P_Z X)^{-1}\), where \(\hat \sigma^2 = \frac{1}{n-{\rm dim}(\beta)} \sum_{i=1}^{n} e^2_i\) with \(e_i = y_i -X\hat \beta^{TSLS}\)

• The Wald estimator is the special case of the instrumental variable estimator where the instrument is a binary variable and therefore defines two groups (\(z = 0\) and \(z=1\)).
• The Wald estimate is then \[\hat \beta_2 = \frac{\bar y_1 - \bar y_0}{\bar x_1 - \bar x_0}\]

• Africa experienced poor economic performances during the second half of the twentieth century, which can be explained by its experience of slave trade and colonialism

• In particular, slave trade may induce long-term negative effects on the economic development of African countries because of induced corruption, ethnic fragmentation and weakening of established states

• Africa experienced, between 1400 and 1900 four slave trades: the trans-Atlantic slave trade (the most important), but also the trans-Saharan, the Red Sea and the Indian Ocean slave trades.

• Not including those who died during the slave trade process, about 18 millions slaves were exported from Africa

• Nunn (2008) conducted a quantitative analysis of the effects of slave trade on economic performances, by regressing the 2000 GDP per capita of 52 African countries on a measure of the level of slave extraction.

• The slave_trade data set is available in the necountries R package
• The response is gdp and the main covariate is a measure of the level of slave extraction, which is the number of slaves normalized by the area of the country
• Let’s explore a scatterplot, using log scales for both variables, which clearly indicates a negative relationship between slaves extraction and per capita GDP in 2000

• Nunn (2008) presents a series of linear regressions, with different sets of controls
• We must work with the covariate colony: grouping the levels having low observations

• We just consider Nunn’s first specification, which includes only dummies for the colonizer as supplementary covariates
• The coefficient is negative and highly significant: it implies that a 10% increase of (log) slave extraction induces a reduction of 1% of (log) GDP per capita

• As noticed by Nunn (2008), the estimation of the effect of slave trade on GDP can be inconsistent for two reasons:

1. the level of slave extraction, which is based on information of the ethnicity of individual slaves and then aggregated at the current countries’ level can be prone to error of measurement; moreover, for countries inside the continent (compared to coastal countries), a lot of slaves died during the journey to the coastal port of export, so the level of extraction may be underestimated for these countries
2. the average economic conditions may be different for countries that suffered a large extraction, compared to the others; in particular, if countries where the trade was particularly important were poor, their current poor economic conditions can be explained by their poor economic conditions 600 years ago and not by slave trades

1. Measurement error induces an attenuation bias, which means that without measurement error, the negative effect of slave trades on GDP per capita would be stronger
2. The second effect would induce an upward-bias (in absolute value) of the coefficient on slave trades

• But, actually, Nunn (2008) showed that areas of Africa that suffered the most slave trade were in general not the poorest areas, but the most developed ones
  
• In this case, the OLS estimator would underestimate the effect of slave trades on GDP per capita
• Nunn (2008) then performs instrumental variable regressions, using as instruments the distance between the centroid of the countries and the closest major market for the four slave trades (for example Mauritius and Oman for the Indian Ocean slave trade and Massawa, Suakin and Djibouti for the Red Sea slave trade)

• The IV regression can be performed by first regressing the endogenous covariate on the external instruments (atlantic, indian, redsea and sahara) and on the exogenous covariates (here colony, the factor indicating the previous colonizer)

• The 4 instruments and the exogenous covariate explain more than 30% of the variance of slave extraction; the F-test just suggest a moderate improvement of considering covariates for explaining the slave extraction
  

• The second stage is obtained by regressing the response on the fitted values of the first-step estimation
• The coefficient has almost doubled, compared to the OLS estimator, which confirms that this latter estimator is biased, with an attenuation bias due to measurement error and a downward-bias caused by the fact that the most developed African regions were more affected by slave trade

• Same results must be extracted via the ivreg() function

• Ananat, Elizabeth Oltmans (2011). “The Wrong Side(s) of the Tracks: The Causal Effects of Racial Segregation on Urban Poverty and Inequality.” American Economic Journal: Applied Economics 3 (2): 34–66.
• Nunn, Nathan. 2008. “The Long-Term Effects of Africa’s Slave Trades.” The Quarterly Journal of Economics 123 (1): 139–76.

• Instruments should be not only uncorrelated with the error of the model, but they should also be correlated with the covariates

• Therefore, if the correlation between the endogenous variable and the instruments is weak, i.e., if the IV is performed using weak instruments, the estimator will not only be highly imprecise, but it will also be seriously biased, in the direction of the OLS estimator

• While performing IV estimation it is therefore important to check that the instruments are sufficiently correlated with the endogenous covariate

• This can be performed using an F test for the first stage regression, comparing the fit for the regression of the endogenous covariates on the set of the exogenous covariates, and on the external instruments.

• A rule of thumb often used is that the F statistic should be at least equal to 10 (a less strict rule is at least equal to 5)

• The F statistic of the first stage regression is the same reported in the weak instruments raw
• In such a case (\(F>10\) and p-value extremely low), we can conclude that the instrument is not weak

• Here, the problem is a bit more complex, because we are considering in the main regression the regressor colony, which is also considered as an instrument
• In such a case, let

1. \(RSS_r\)= residual sum of square of the regression of (log) slaveareas against colony (having in total 5 parameters)
2. \(RSS_u\)= residual sum of square of the regression of (log) slaveareas against colony and the remaining instruments (having in total 9 parameters)

• The \(F\) statistic is then \[F=\frac{(RSS_r−RSSu)/q}{RSSu/(n−k)}\] where \(q = 4\) (the number of instruments excluded in the reduced model) and \(n-k = 52 - 9 = 43\)
  
• Although the instruments are statistically significant (p = 0.0024), the first-stage \(F = 4.89\) is low, indicating that the instruments are weak (they are not unuseful but weak, meaning that they are not generating an IV reliable)

• The Hausman test evaluates whether regressors treated as endogenous actually are endogenous. It compares:

1. IV/2SLS estimates (consistent even if regressors are endogenous)

2. OLS estimates (more efficient but inconsistent if regressors are endogenous)

• If the two sets of estimates differ significantly there is evidence of endogeneity.

• Suppose the structural model is \[y = X\beta + \varepsilon\]

• Some regressors in \(X\) may be endogenous, then let’s denote the two estimators with

1. \(\hat{\beta}_{\text{OLS}}\): OLS estimator,
2. \(\hat{\beta}_{\text{IV}}\) IV/2SLS estimator.

• The Hausman statistic is \[H = (\hat{\beta}_{\text{IV}} - \hat{\beta}_{\text{OLS}})' \left[ \operatorname{Var}(\hat{\beta}_{\text{IV}}) - \operatorname{Var}(\hat{\beta}_{\text{OLS}}) \right]^{-1} (\hat{\beta}_{\text{IV}} - \hat{\beta}_{\text{OLS}})\]

• Under the null hypothesis \(H_0\), the regressors are exogenous, so OLS and IV are both consistent and the difference between the estimators should be close to zero

• The ivreg() reports a different test called Wu-Hausman, which is obtained in a slightly different way

1. For the first example (on segregation): the instruments are not weak but we do not need to use them, because the Wu-Hausman does not reject exogeneity \(\implies\) OLS preferred to IV, because it is more efficient and unbiased
2. For the second example (on slave trade): the instruments are not weak and we have moderately evidence to use them, because the Wu-Hausman test (“partially”) reject the exogeneity; it is plausible to consider the IV regression