# Covariance Matrix Estimation for the Generalized Method of Moments

I am solving and empirical exercise on the Generalized Method of Moments. It's a classical application/test of a famous model in Economics. There are 2 parameters $$(\beta, \gamma)$$ to be estimated with 5 moment conditions. The theoretical moment conditions are:

$$E\left(\left[\beta \left(\frac{c_{t+1}}{c_t}\right)^{-\gamma}R_{t+1} - 1\right]\otimes z_{t}\right) = 0, \qquad g(data_t; \beta, \gamma)\equiv \beta \left[\left(\frac{c_{t+1}}{c_t}\right)^{-\gamma}R_{t+1} - 1\right]\otimes z_{t}$$

where $$c_{t}$$ denotes the household consumption, $$R_t$$ is the return on the SP500 index and $$z_t = (1, R_{t}, R_{t-1}, \frac{c_{t}}{c_{t-1}}, \frac{c_{t-1}}{c_{t-2}})$$ is a vector of instruments. The goal is estimating the parameters using a two-stage GMM procedure: get a first estimate using and identity matrix, then use this estimate to compute the efficient weighting matrix that will lead to the final estimate. I will leave the data I am using posted below using dput. I am using R on my computation. We are not supposed to use packages, we must implement the optimization on our own.

I get sensible estimates but the final covariance matrix for the estimators isn't invertible, which smells very fishy. The code is below:

moments <- function(params){
beta <- params[1]; g <- params[2];
m1 <- beta * gmm.data$$c_ratio^(-g) * gmm.data$$R - 1
m2 <- m1 * gmm.data$$R_lag1 m3 <- m1 * gmm.data$$R_lag2
m4 <- m1 * gmm.data$$c_ratio_lag1 m5 <- m1 * gmm.data$$c_ratio_lag2
return(do.call(cbind, list(m1, m2, m3, m4, m5)))}

# Using the identity matrix for estimation
gmm.fn <- function(x){colMeans(moments(x)) %*% colMeans(moments(x))}
first.pass <- optim(par = c(0.8, 2), gmm.fn, lower = c(0, 0), upper = c(0.9999, 200))

m <- moments(first.pass$par) # Computing the HC-only covariance matrix for moments S.hat <- 1/nrow(gmm.data) * t(m) %*% m W.hc <- solve(S.hat) gmm.fn.hc <- function(x) {colMeans(moments(x)) %*% W.hc %*% colMeans(moments(x))} second.pass.hc <- optim(par = first.pass$par, gmm.fn.hc, lower = c(0, 0), upper = c(1, 200))


The code above will give the estimates for the parameters. They are around what would be expected. To compute the covariance matrix, we need to approximate the Jacobian matrix. Let $$\theta = (\beta, \gamma)'$$. Then, we must compute

$$G = E\left(\frac{\partial g(data_t; \theta)}{\partial \theta'}\right)$$

It turns out that the Kroencker structure helps a bit since $$\frac{\partial g(data_t; \theta)}{\partial \theta'} = \begin{bmatrix} \left[\left(\frac{c_{t+1}}{c_t}\right)^{-\gamma}R_{t+1}\right]\otimes z_{t}\\ -\beta\left[\left(\frac{c_{t+1}}{c_t}\right)^{-\gamma}R_{t+1}\cdot \log(c_{t+1}/c_t)\right]\otimes z_{t} \end{bmatrix}$$

I implemented the following function to use the common approximation:

# Compute the empirical G matrix
beta <- params[1]; g <- params[2];

# Derivatives with respect to beta
db1 <- mean(gmm.data$$c_ratio^(-g) * gmm.data$$R)
db2 <- mean(db1 * gmm.data$$R_lag1) db3 <- mean(db1 * gmm.data$$R_lag2)
db4 <- mean(db1 * gmm.data$$c_ratio_lag1) db5 <- mean(db1 * gmm.data$$c_ratio_lag2)
db <- c(db1, db2, db3, db4, db5)

# Derivatives with respect to gamma
dg1 <- mean(-beta * gmm.data$$c_ratio^(-g) * gmm.data$$R * log(gmm.data$$c_ratio)) dg2 <- mean(dg1 * gmm.data$$R_lag1)
dg3 <- mean(dg1 * gmm.data$$R_lag2) dg4 <- mean(dg1 * gmm.data$$c_ratio_lag1)
dg5 <- mean(dg1 * gmm.data\$c_ratio_lag2)
dg <- c(dg1, dg2, dg3, dg4, dg5)
return(rbind(db, dg))
}


The final covariance matrix is $$(G'S^{-1}G)^{-1}$$. Just notice that, technically, the Grad function is really computing $$G'$$. Now, my problem:

HC.cov.mat <- solve(Grad(second.pass.hc$$par) %*% solve(S.hat) %*% t(Grad(second.pass.hc$$par)))


R is complaining that HC.cov.mat cannot be computed because it cannot invert $$(G'S^{-1}G)$$. I can't understand what is wrong either with the code of the derivations. Any ideas?

The dataset:

> dput(head(gmm.data, 150))
structure(list(pcecc96 = c(1976.01403808594, 1994.91796875, 2020.08203125,
2012.00805664062, 2014.61401367188, 2013.86499023438, 2043.8349609375,
2053.77392578125, 2095.083984375, 2117.27709960938, 2143.30590820312,
2160.580078125, 2191.14990234375, 2206.4990234375, 2227.32006835938,
2257.53588867188, 2276.34692382812, 2321.02392578125, 2362.0810546875,
2405.712890625, 2412.580078125, 2466.38208007812, 2493.583984375,
2536.42993164062, 2607.73608398438, 2646.23291015625, 2653.01391601562,
2683.30908203125, 2694.48803710938, 2710.01391601562, 2747.06396484375,
2761.12890625, 2778.02587890625, 2844.60107421875, 2887.9169921875,
2942.0419921875, 2955.31298828125, 2988.08203125, 3007.35791015625,
3022.02001953125, 3046.17309570312, 3065.07397460938, 3078.958984375,
3105.95190429688, 3097.52001953125, 3156.97802734375, 3185.99291992188,
3211.39404296875, 3264.67895507812, 3307.76708984375, 3370.73608398438,
3422.68188476562, 3503.0009765625, 3566.95703125, 3565.34912109375,
3577.89794921875, 3567.22705078125, 3535.31201171875, 3548.041015625,
3563.28198242188, 3511.19702148438, 3540.64697265625, 3598.88598632812,
3650.03491210938, 3689.28491210938, 3763.04809570312, 3797.669921875,
3837.65405273438, 3887.44702148438, 3933.26806640625, 3954.59594726562,
3991.98901367188, 4052.01000976562, 4074.77392578125, 4161.88720703125,
4179.36279296875, 4213.087890625, 4234.9091796875, 4232.1591796875,
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4873.1650390625, 4936.29296875, 5020.1630859375, 5066.25, 5162.47509765625,
5173.64306640625, 5218.85107421875, 5275.72021484375, 5369.048828125,
5401.97998046875, 5407.3720703125, 5481.166015625, 5543.72314453125,
5555.451171875, 5653.587890625, 5695.326171875, 5745.93310546875,
5811.26416015625, 5838.22509765625, 5865.47705078125, 5922.26123046875,
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6161.39794921875, 6203.22412109375, 6269.7177734375, 6344.44580078125,
6368.77197265625, 6426.703125, 6498.22900390625, 6555.27392578125,
6630.2529296875, 6681.8408203125, 6732.791015625, 6805.5810546875,
6822.51904296875, 6882.330078125, 6944.7001953125, 6993.14404296875,
7057.64111328125, 7133.56591796875, 7176.75390625, 7233.91796875,
7310.169921875), sp500 = c(57.7633018493652, 56.2766990661621,
56.0699996948242, 55.7200012207031, 55.3333015441895, 62.0032997131348,
65.9832992553711, 66.8300018310547, 70.2733001708984, 69.8600006103516,
62.2233009338379, 57.8300018310547, 59.61669921875, 65.5500030517578,
69.6699981689453, 70.9666976928711, 73.2733001708984, 77.5466995239258,
80.3000030517578, 82.8767013549805, 84.75, 86.5667037963867,
87.4300003051758, 86.9266967773438, 91.7566986083984, 91.629997253418,
88.1466979980469, 81.4332962036133, 79.8167037963867, 87.0766983032227,
91.6600036621094, 94.4366989135742, 94.5400009155273, 91.6267013549805,
98.013298034668, 99.9032974243164, 105.233299255371, 100.933296203613,
101.680000305176, 94.4666976928711, 94.2799987792969, 88.7067031860352,
79.1999969482422, 78.7399978637695, 86.2332992553711, 96.7332992553711,
101.440002441406, 98.5466995239258, 96.4132995605469, 105.400001525879,
108.166702270508, 109.199996948242, 114.066703796387, 115, 107.433296203613,
105.066703796387, 102.193298339844, 95.6667022705078, 90.6399993896484,
74.486701965332, 69.4167022705078, 78.8133010864258, 89.0733032226562,
87.6232986450195, 89.1132965087891, 99.5199966430664, 101.63330078125,
104.333297729492, 102.599998474121, 101.800003051758, 99.0333023071289,
98.0599975585938, 93.9467010498047, 89.3499984741211, 95.9266967773438,
101.663299560547, 97.1399993896484, 99.3467025756836, 101.176696777344,
106.233299255371, 105.333297729492, 110.300003051758, 108.433296203613,
123.266700744629, 133.13330078125, 131.533294677734, 132.800003051758,
125.666702270508, 122.166702270508, 114.199996948242, 114.13330078125,
113.833297729492, 136.733306884766, 147.666702270508, 162.733306884766,
165.533294677734, 165.766693115234, 160.36669921875, 155.766693115234,
160.533294677734, 165.199996948242, 177.300003051758, 184.800003051758,
188.300003051758, 197, 219.966705322266, 240.600006103516, 241.166702270508,
243.699996948242, 279.299987792969, 293.266693115234, 319.399993896484,
255.399993896484, 258.100006103516, 263.13330078125, 266.933288574219,
274.966705322266, 290.700012207031, 313.299987792969, 341.933288574219,
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317.053314208984, 353.343292236328, 378.653289794922, 385.609985351562,
387.103302001953, 412, 410.163299560547, 417.153289794922, 423.660003662109,
442.363311767578, 445.463287353516, 453.553314208984, 464.246704101562,
469.459991455078, 450.986694335938, 460.86669921875, 460.003295898438,
480.106689453125, 523.690002441406, 565.083312988281, 597.673278808594,
637.010009765625, 658.966674804688, 660.543273925781, 726.793273925781,
785.590026855469), R = c(0.98353996764165, 0.974263888392671,
0.996327087857536, 0.993757829926412, 0.993059948527604, 1.12054220483516,
1.06419012472966, 1.01283207395263, 1.05152324174026, 0.994118682920231,
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1.06285270671817, 1.01861202178851, 1.03250260408072, 1.05832137140077,
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1.00997261615528, 0.994243354385506, 1.05556407881719, 0.99861915961557,
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1.00739799580166, 0.929058786480575, 0.998023653645847, 0.940885705712529,
0.892829900150211, 0.99419193052781, 1.09516512058542, 1.12176270756967,
1.04865649390919, 0.971477692745998, 0.978351380881498, 1.09321018994572,
1.02624953230147, 1.00955279819061, 1.04456691377429, 1.00818202133095,
0.934202575683594, 0.977971518227075, 0.972651607476794, 0.936134793813663,
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1.10203115788865, 1.01720599087285, 1.00140997880792, 0.967424132103965,
0.971315702537215, 1.03060090361534, 1.02906999622649, 1.07324459035739,
1.04230118370506, 1.01893939362663, 1.04620285080851, 1.11658226051911,
1.09380192675533, 1.00235534560522, 1.01050433021592, 1.14608121169689,
1.05000610788647, 1.08911104259283, 0.799624291725121, 1.01057170035848,
1.01950133498143, 1.01444130325461, 1.03009522263393, 1.0572189526231,
1.07774329080469, 1.09139260101143, 1.01013854292082, 0.973634367069925,
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0.995541989224628, 1.01704196899592, 1.01559789656792, 1.0441469762163,
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0.960649901045064, 1.02190753076953, 0.998126566050929, 1.0437027163369,
1.0907783914403, 1.07904162835628, 1.05767285119069, 1.06581644579367,
1.03446832028141, 1.00239253240167, 1.10029622980832, 1.08089886772355
), R_lag1 = c(1.02127232067638, 0.98353996764165, 0.974263888392671,
0.996327087857536, 0.993757829926412, 0.993059948527604, 1.12054220483516,
1.06419012472966, 1.01283207395263, 1.05152324174026, 0.994118682920231,
0.890685662613892, 0.929394631322202, 1.03089568271007, 1.09952419222736,
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