# pedroni99m: panel cointegration tests - multivatiate case In pco: Panel Cointegration Tests

## Description

Computation of the Pedroni (1999) panel cointegration test statistics. All statistics are asymptotically normal. Reported are their empirical values and their standardized values (as suggested in Pedroni, 1999).

## Usage

 `1` ```pedroni99m(X, kk = 0, type.stat = 1, ka = 2) ```

## Arguments

 `X` The data to be tested for cointegration. Must be a 'cube', an array (TxNxM) with multiple 'sheets', the first 'sheet' is the 'dependent' variable, 'independent' variables are the rest. The first dimension is 'time', the second is 'individuals' and the third is 'variables'. No missing values are allowed. `kk` Parameter for the Newey-West (1994) long term variance estimation (number of lags). Can be a vector, with a different value for each individual series, or a scalar. By default it is set to 'round(4 * (T/100)^(2/9))'. `type.stat` Type of the main regresion: 1 - 'none', 2 - 'intercept', 3 - 'intercept and time trend'. `ka` Number of lags for the ADF type regression on residuals, for the parametric statistics.

## Details

The function closely follows the instructions in Pedroni (1999). Calculated and reported are the 7 statistics on page 660 in Pedroni (1999) for the multivariate case. Also reported are their standardized values, as described on page 665 and by use of the adjustment terms in Table 2, page 666, op.cit. H0 is 'no cointegration'.

## Value

 `CALL` The result of 'match.call()'. `METHOD` Title of the test. `STATISTIC` The 7 test statistics in Pedroni (1999), in two columns - for the empirical and the standardized values.

## Note

Under H0 ('no cointegration') the autoregressive coefficients, gamma_i = 1 for all i, versus H1: gamma_i < 1 for all i.

The standardized values of the test statistics are asymptotically normal (0,1) under H0.

Georgi Marinov

## References

Newey, Whitney K.; West, Kenneth D. (1994). "Automatic lag selection in covariance matrix estimation". Review of Economic Studies 61 (4): 631-654.

Pedroni, Peter, 1999. "Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 61(0), pages 653-70, Special I.

`pedroni99`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227``` ```xx<-array(cumsum(rnorm(10000)),dim=c(100,20,5)) pedroni99m(xx) ## The function is currently defined as function (X, kk = 0, type.stat = 1, ka = 2) { ffm <- function(Y2, X2) { NN = ncol(X2) sapply(1:NN, function(l) { lm(Y2[, l] ~ X2[, l, ] - 1)\$residuals }) } ff1m <- function(Y2, X2) { NN = ncol(X2) sapply(1:NN, function(l) { lm(Y2[, l] ~ X2[, l, ])\$residuals }) } ff2m <- function(Y2, X2) { NN = ncol(X2) trend = 1:nrow(X2) sapply(1:NN, function(l) { lm(Y2[, l] ~ X2[, l, ] + trend)\$residuals }) } ffmm <- function(Y1, X1) { NN = ncol(X1) sapply(1:NN, function(l) { lm(Y1[, l] ~ X1[, l] - 1)\$residuals }) } nwm <- function(xx, ki) { tt = length(xx) (1/tt) * sum(sapply(1:ki, function(s) { (1 - s/(ki + 1)) * sum(xx[(s + 1):tt] * xx[1:(tt - s)]) })) } adflm<-function (ee, lags) { nn<-length(ee) z<-ee[(lags+1):nn] zl<-ee[lags:(nn-1)] zd<-matrix(cbind(rep(z,lags)),ncol=lags) ii<-embed(1:nn,lags) ii<-ii[-(nrow(ii)),] zd<-zd-ee[ii] zd<-zd[,-1] z<-ee[(lags+1):nn] zl<-ee[lags:(nn-1)] return(lm(z ~ zl + zd -1)\$residuals) } na.fail(X) Y <- as.matrix(X[, , 1]) XX <- X[, , (2:dim(X)[3])] TD <- dim(X)[1] N <- dim(X)[2] M <- dim(X)[3] if (is.vector(kk) && length(kk) == N) { k = kk } else if (kk > 0) { k = rep(round(kk), N) } else { i = round(4 * (TD/100)^(2/9)) k = rep(i, N) } if (ka < 2) { ka = 2 warning("Parameter 'ka' was changed to 2.") } ka <- as.vector(ka) if (length(ka) != N) { ka <- rep(ka[1], N) } stamm <- array(dim = c(7, 3, 6)) stamm[, , 1] <- cbind(c(6.982, -6.388, -1.662, -1.662, -9.889, -1.992, -1.992), c(11.754, -9.495, -2.177, -2.177, -12.938, -2.453, -2.453), c(21.162, -14.011, -2.648, -2.648, -17.359, -2.872, -2.872)) stamm[, , 2] <- cbind(c(10.402, -10.191, -2.156, -2.156, -13.865, -2.44, -2.44), c(15.197, -13.256, -2.567, -2.567, -16.888, -2.827, -2.827), c(24.556, -17.6, -2.967, -2.967, -21.116, -3.179, -3.179)) stamm[, , 3] <- cbind(c(14.254, -14.136, -2.571, -2.571, -17.834, -2.819, -2.819), c(18.91, -17.163, -2.93, -2.93, -20.841, -3.157, -3.157), c(28.046, -21.287, -3.262, -3.262, -24.93, -3.464, -3.464)) stamm[, , 4] <- cbind(c(18.198, -18.042, -2.926, -2.926, -21.805, -3.151, -3.151), c(22.715, -21.013, -3.241, -3.241, -24.775, -3.452, -3.452), c(31.738, -25.13, -3.545, -3.545, -28.849, -3.737, -3.737)) stamm[, , 5] <- cbind(c(22.169, -21.985, -3.244, -3.244, -25.75, -3.45, -3.45), c(26.603, -24.944, -3.531, -3.531, -28.72, -3.726, -3.726), c(35.537, -28.981, -3.806, -3.806, -32.716, -3.986, -3.986)) stamm[, , 6] <- cbind(c(26.12, -25.889, -3.533, -3.533, -29.627, -3.723, -3.723), c(30.457, -28.795, -3.795, -3.795, -32.538, -3.976, -3.976), c(39.231, -32.756, -4.047, -4.047, -36.494, -4.217, -4.217)) rownames(stamm) <- c("nipanel", "rhopanel", "tpanelnonpar", "tpanelpar", "rhogroup", "tgroupnonpar", "tgrouppar") colnames(stamm) <- c("none", "intercept", "trend") stavv <- array(dim = c(7, 3, 6)) stavv[, , 1] <- cbind(c(81.145, 64.288, 1.559, 1.559, 41.943, 0.649, 0.649), c(104.546, 57.61, 0.964, 0.964, 51.49, 0.618, 0.618), c(160.249, 64.219, 0.69, 0.69, 66.387, 0.555, 0.555)) stavv[, , 2] <- cbind(c(140.804, 89.962, 1.286, 1.286, 57.801, 0.6, 0.6), c(151.094, 81.772, 0.923, 0.923, 67.123, 0.585, 0.585), c(198.167, 83.815, 0.686, 0.686, 81.832, 0.548, 0.548)) stavv[, , 3] <- cbind(c(182.45, 103.176, 1.028, 1.028, 72.097, 0.567, 0.567), c(190.661, 99.331, 0.843, 0.843, 81.835, 0.56, 0.56), c(239.425, 103.905, 0.688, 0.688, 97.362, 0.543, 0.543)) stavv[, , 4] <- cbind(c(217.784, 120.787, 0.928, 0.928, 88.611, 0.559, 0.559), c(231.864, 119.546, 0.8, 0.8, 98.278, 0.553, 0.553), c(276.997, 124.613, 0.686, 0.686, 113.145, 0.538, 0.538)) stavv[, , 5] <- cbind(c(256.53, 132.499, 0.82, 0.82, 103.371, 0.544, 0.544), c(270.451, 134.341, 0.75, 0.75, 113.131, 0.542, 0.542), c(310.982, 138.227, 0.654, 0.654, 127.989, 0.53, 0.53)) stavv[, , 6] <- cbind(c(277.429, 143.561, 0.75, 0.75, 117.059, 0.53, 0.53), c(293.431, 144.615, 0.685, 0.685, 126.059, 0.525, 0.525), c(348.217, 154.378, 0.638, 0.638, 140.756, 0.518, 0.518)) rownames(stavv) <- c("nipanel", "rhopanel", "tpanelnonpar", "tpanelpar", "rhogroup", "tgroupnonpar", "tgrouppar") colnames(stavv) <- c("none", "intercept", "trend") statsm <- matrix(nrow = 7, ncol = 2) rownames(statsm) <- c("nipanel", "rhopanel", "tpanelnonpar", "tpanelpar", "rhogroup", "tgroupnonpar", "tgrouppar") colnames(statsm) <- c("empirical", "standardized") e <- matrix(ncol = N, nrow = TD) if (type.stat == 2) { e <- ff1m(Y, XX) } else if (type.stat == 3) { e <- ff2m(Y, XX) } else { e <- ffm(Y, XX) type.stat = 1 } De <- diff(e) estar <- e Destar <- diff(estar) DXX <- array(dim = c((dim(XX)[1] - 1), dim(XX)[2], dim(XX)[3])) DXX[, , 1:dim(XX)[3]] <- sapply(1:dim(XX)[3], function(i) { DXX[, , i] <- diff(XX[, , i]) }) DY <- diff(Y) eta <- ffm(DY, DXX) L11hat2 <- sapply(1:N, function(i) { (1/nrow(eta)) * sum(eta[, i]^2) + 2 * nwm(eta[, i], k[i]) }) mu <- matrix(ncol = ncol(DY), nrow = nrow(DY)) mu <- ffmm(e[2:TD, ], e[1:(TD - 1), ]) lambdahat <- sapply(1:N, function(i) { nwm(mu[, i], k[i]) }) mustar <- matrix(ncol = ncol(DY), nrow = nrow(DY)) mustar <- sapply(1:N, function(i) { adflm(e[, i], ka[i]) }) shatstar2 <- sapply(1:N, function(i) { (1/nrow(mustar)) * sum(mustar[, i]^2) }) stildestar2 <- (1/N) * sum(shatstar2) shat2 <- sapply(1:N, function(i) { (1/nrow(mu)) * sum(mu[, i]^2) }) sigmahat2 <- shat2 + 2 * lambdahat sigmatilde2 <- (1/N) * sum(L11hat2^(-2) * sigmahat2) nipa <- sum(sapply(1:N, function(i) { sum((L11hat2[i]^(-2)) * (e[1:(TD - 1), i]^2)) })) lel <- sum(sapply(1:N, function(i) { (L11hat2[i]^(-2)) * sum(sapply(2:(nrow(De)), function(ttt) { (e[(ttt - 1), i] * De[ttt, i] - lambdahat[i]) })) })) nipanel <- (TD^2) * (N^(3/2)) * nipa^(-1) statsm[1, 1] <- nipanel rhopanel <- TD * (N^(1/2)) * (nipa^(-1)) * lel statsm[2, 1] <- rhopanel tpanelnonpar <- ((sigmatilde2 * nipa)^(-1/2)) * lel statsm[3, 1] <- tpanelnonpar tpanelpar <- ((stildestar2 * sum(sapply(1:N, function(i) { sum((L11hat2[i]^(-2)) * estar[1:(nrow(estar) - 1), i]^2) })))^(-1/2)) * sum(sapply(1:N, function(i) { sum(sapply(2:(nrow(Destar)), function(ttt) { (L11hat2[i]^(-2)) * (estar[(ttt - 1), i] * Destar[ttt, i]) })) })) statsm[4, 1] <- tpanelpar rhogroup <- TD * (N^(-1/2)) * sum(sapply(1:N, function(i) { ((sum(e[1:(nrow(e) - 1), i]^2))^(-1)) * sum(sapply(2:(nrow(De)), function(ttt) { (e[(ttt - 1), i] * De[ttt, i] - lambdahat[i]) })) })) statsm[5, 1] <- rhogroup tgroupnonpar <- (N^(-1/2)) * sum(sapply(1:N, function(i) { ((sigmahat2[i] * sum(e[1:(nrow(e) - 1), i]^2))^(-1/2)) * sum(sapply(2:(nrow(De)), function(ttt) { (e[(ttt - 1), i] * De[ttt, i] - lambdahat[i]) })) })) statsm[6, 1] <- tgroupnonpar tgrouppar <- (N^(-1/2)) * sum(sapply(1:N, function(i) { (sum(shat2[i] * estar[1:(nrow(estar) - 1), i]^2))^(-1/2) * sum(sapply(2:nrow(Destar), function(tt1) { estar[(tt1 - 1), i] * Destar[tt1, i] })) })) statsm[7, 1] <- tgrouppar statsm[, 2] <- sapply(1:7, function(i) { (statsm[i, 1] - stamm[i, type.stat, M] * sqrt(N))/sqrt(stavv[i, type.stat, M]) }) list(CALL = match.call(), METHOD = "Pedroni(1999) panel tests for cointegration", STATISTIC = statsm) } ```