# Test on SUR model

I have a SUR model with 22 equations, where each equation has the same 7 factors. I want to test if a coefficient (b3 in equation 1) is significantly different from another coefficient in another equation (for example, b3 in equation 11) of the SUR model.

• Can you clarify your question? Eg, I don't see a "?" anywhere. Are you asking how this can be done? – gung - Reinstate Monica Jan 27 '16 at 0:14
• acronyms should also be spelled out the first time used. I'm guessing SUR is seemingly unrelated regressions? – StatsStudent Jan 27 '16 at 0:23
• i'm sorry yes SUR is Seemingly Unrelates Regression and yes i would know how could i do the test – a.paolini Jan 27 '16 at 10:44

Conceptually, there is not all that much of a difference to any other $t$ or Wald type test. Think of the SUR estimates of the 22 equations being stacked on top of each other in one long $22\cdot 7$ (plus possibly estimates for the constant term) vector $\hat\delta$.

As you can read up for example very nicely in Hayashi, Econometrics, $\hat\delta$ will be asymptotically normal provided the SUR assumptions (the most important of which is that your regressors are actually predetermined) are met,

$$\sqrt{n}(\hat\delta-\delta)\to_dN(0,Avar(\hat\delta))$$ Then, you may test hypotheses like $$H_{0}:\delta_{\ell}=\bar{\delta}_{\ell}$$ with the $t$-statistic $$t_{\ell}=\frac{\sqrt{n}(\widehat{\delta}_{\ell}-\bar{\delta}_{\ell})}{\sqrt{\widehat{\mathrm{Avar}} (\widehat{\delta}_{\ell})}}$$ where $\widehat{\mathrm{Avar}}(\widehat{\delta}_{\ell})$ is the $(\ell,\ell)$-element of the estimated version of $Avar(\hat\delta)$. Similarly, test $$H_{0}:R\delta =r$$ (with $J$ rows for the $J$ hypotheses) with the Wald statistic $$W=n(R\widehat{\delta}-r)'\left[R\widehat{\mathrm{Avar}}(\widehat{\delta})R'\right]^{-1}(R\widehat{\delta}-r).$$ These test statistics then follow their usual asymptotic $N(0,1)$ and $\chi^2_J$ null distributions.

The SUR estimator for $Avar(\hat\delta)$ is given by

$$\widehat{Avar}(\widehat{\delta})=\left(% \begin{array}{ccc} \widehat{\sigma}^{11}\frac{1}{n}\sum_{i=1}^nz_{i1}z_{i1}' & \cdots & \widehat{\sigma}^{1M}\frac{1}{n}\sum_{i=1}^nz_{i1}z_{iM}' \\ \vdots & \ddots &\vdots \\ \widehat{\sigma}^{M1}\frac{1}{n}\sum_{i=1}^nz_{iM}z_{i1}' & \cdots & \widehat{\sigma}^{MM}\frac{1}{n}\sum_{i=1}^nz_{iM}z_{iM}' \\ \end{array} \right)^{-1},$$ with $z_{im}$ the regressors of equation $m$ and $\widehat{\sigma}^{mk}$ the estimate of the covariance of the error terms of equations $m$ and $k$, using preliminary (OLS) residuals to estimate these.

So for your problem, you would just need to specify a row vector $R$ that has a $1$ and a $-1$ in the appropriate places corresponding to your coefficients of interest, as well as $r=0$, to test coefficient equality.

Here is an example slightly adapted from the systemfit package that should adaptable to your problem:

library(systemfit)
data("Kmenta")
eqDemand <- consump ~ price + income
eqSupply <- consump ~ price + farmPrice + trend
system <- list(demand = eqDemand, supply = eqSupply)
fitsur <- systemfit(system, "SUR", data = Kmenta)
print(fitsur)

Rmat <- matrix(0, nrow = 1, ncol = 7)
Rmat[1, 2] <- 1
Rmat[1, 5] <- -1
qvec <- 0
linearHypothesis(fitsur, Rmat, test = "Chisq")


Output:

> linearHypothesis(fitsur, Rmat, test = "Chisq")
Linear hypothesis test (Chi^2 statistic of a Wald test)

Hypothesis:
demand_price - supply_price = 0

Model 1: restricted model
Model 2: fitsur

Res.Df Df  Chisq Pr(>Chisq)
1     34
2     33  1 31.712  1.788e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


As an aside, if each of your equations has the same regressors, SUR just boils down to equation-by-equation OLS.