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\begin{document}

\title{Supplementary Material to `Estimation of spatial autoregressions with stochastic weight matrices'\footnote{This appendix should be read in conjunction with Section 6 of `Estimation of spatial autoregressions with stochastic weight matrices'.}}
\date{\today}
\author{Abhimanyu Gupta\thanks{\emph{Email}: a.gupta@essex.ac.uk. }\\ Department of Economics\\ University of Essex}
\maketitle
\section*{A simulation study when spatial weights are generated by dependent random variables}\label{subsec:MC_dep}
Section 6.1 of `Estimation of spatial autoregressions with stochastic weight matrices' took the spatial weights to be iid but in some cases, especially for asymmetric spatial weight matrices, this may not be reasonable. For instance, if the distance from unit $r$ to $s$ is small the distance from $s$ to $r$ may also be expected to be small. To capture such behaviour we use the same designs as described in Section 6.1, but with the following alteration: after generating the $V_j$, replace $v_{rs,j}=\left(v_{sr,j}^2+5\right)^{\frac{1}{2}}$ for each $r=1,\ldots,m$ and $s\leq r$, where $v_{rs,j}$ denotes the $(r,s)$-th element of $V_j$. Thus we replace the part of $V_j$ below the diagonal with a transformation of the part above the diagonal. The choice of transformation is uniformly continuous, in keeping with the idea of `preserving' the distance between units discussed earlier in the paragraph. Similar operations are carried out with the sparse and circulant specifications of $W$. We then proceed with the experiment design as in the corresponding parts of Section 6.1.

The results are in Tables \ref{table_dep_bias_MSE_stoch_t}(a)-(c), where we report the stochastic case. They indicate that the procedure of generating dependent weights in this way does little to alter the character and behaviour of the estimates. The same features that we saw in Tables 6.1, 6.3 and 6.5 are evident. We may also compare the dense, sparse and circulant cases to see if stochastic dependent spatial weights yield any difference in performance as opposed to stochastic iid ones. Out of 72 comparisons for each type of weight matrix, the dependent setting exhibits a smaller bias in 54 (dense), 61 (sparse) and 41 (circulant) cases, while the MSE is smaller in all 72 (dense), 54 (sparse) and 32 (circulant) cases. Thus in our experiment designs dependent spatial weights do not contaminate the performance of estimates.
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\begin{table}[htp]
\normalsize{
\begin{center}

\begin{tabular}{cccccccccccccccccccccccc}
&&\multicolumn{6}{c}{(a) Dense $W_1,W_2$}&\multicolumn{6}{c}{(b) Sparse $W$}&\multicolumn{6}{c}{(c) Circulant $W$}
\\
\cmidrule{4-7}\cmidrule{10-13}\cmidrule{16-19}
&$m/n$& \multicolumn{2}{c}{48}  &\multicolumn{2}{c}{72} & \multicolumn{2}{c}{144}                             & \multicolumn{2}{c}{96}  &\multicolumn{2}{c}{144} & \multicolumn{2}{c}{288}                    & \multicolumn{2}{c}{96}  &\multicolumn{2}{c}{144} & \multicolumn{2}{c}{288}  \\\midrule
Bias&$v$& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}                              & {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}                      & {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}\\
                                                                                                                                                                                                                 \midrule
$\hat\theta$&    1 & 0.0292 & 0.0025 & 0.0423 & 0.0046 & 0.0459 & 0.0009                                   & 0.0280 & 0.0061 & 0.0585 & 0.0059 & 0.0586 & 0.0038                             & 0.0058 & 0.0309 & 0.0024 & 0.0151 & 0.0027 & 0.0152 \\
                                                                                                                                                                                                                    
  &  10 & 0.0126 & 0.0143 & 0.0059 & 0.0069 & 0.0051 & 0.0046                                            & 0.0017 & 0.0063 & 0.0062 & 0.0062 & 0.0005 & 0.0009                             & 0.0047 & 0.0367 & 0.0017 & 0.0139 & 0.0015 & 0.0128 \\
                                                                                                                                                                                                                   
 &   20 & 0.0086 & 0.0104 & 0.0067 & 0.0104 & 0.0016 & 0.0050                                            & 0.0033 & 0.0151 & 0.0027 & 0.0046 & 0.0001 & 0.0031                            & 0.0066 & 0.0422 & 0.0024 & 0.0099 & 0.0001 & 0.0074 \\
                                                                                                                                                                                                                    
&    100 & 0.0036 & 0.0030 & 0.0023 & 0.0084 & 0.0012 & 0.0013                                           & 0.0100 & 0.0109 & 0.0046 & 0.0076 & 0.0021 & 0.0027                             & 0.0055 & 0.0369 & 0.0047 & 0.0346 & 0.0003 & 0.0044 \\
                                                                                                                                                                                                                    
$\tilde\theta$&    1 & 0.0698 & 0.0151 & 0.0622 & 0.0084 & 0.0793 & 0.0066                                  & 0.0008 & 0.0063 & 0.0396 & 0.0055 & 0.0417 & 0.0039                           & 0.0697 & 0.4591 & 0.0769 & 0.4970 & 0.0874 & 0.5451 \\
                                                                                                                                                                                                                    
  &  10 & 0.0007 & 0.0021 & 0.0016 & 0.0020 & 0.0012 & 0.0009                                          & 0.0091 & 0.0052 & 0.0116 & 0.0107 & 0.0031 & 0.0049                                   & 0.0810 & 0.5066 & 0.0908 & 0.5429 & 0.1008 & 0.5947 \\
                                                                                                                                                                                                                    
   & 20 & 0.0036 & 0.0038 & 0.0017 & 0.0032 & 0.0031 & 0.0005                                          & 0.0125 & 0.0154 & 0.0090 & 0.0116 & 0.0027 & 0.0044                                   & 0.0819 & 0.5032 & 0.0889 & 0.5336 & 0.1000 & 0.5902 \\
                                                                                                                                                                                                                    
  &  100 & 0.0104 & 0.0131 & 0.0052 & 0.0084 & 0.0058 & 0.0050                                          & 0.0207 & 0.0226 & 0.0120 & 0.0117 & 0.0016 & 0.0026                            & 0.0811 & 0.5006 & 0.0896 & 0.5452 & 0.0994 & 0.5836 \\
                                                                                                                                                                                                                    
 $\check\theta$&   1 & 0.0516 & 0.0073 & 0.0625 & 0.0078 & 0.0531 & 0.0029                                  & 0.1230 & 0.0081 & 0.1311 & 0.0076 & 0.1335 & 0.0063                            & 0.0008 & 0.0102 & 0.0034 & 0.0204 & 0.0090 & 0.0548 \\
                                                                                                                                                                                                                    
   & 10 & 0.0155 & 0.0187 & 0.0116 & 0.0137 & 0.0039 & 0.0062                                           & 0.0306 & 0.0280 & 0.0263 & 0.0265 & 0.0107 & 0.0132                            & 0.0048 & 0.0315 & 0.0059 & 0.0339 & 0.0118 & 0.0715 \\
                                                                                                                                                                                                                    
   & 20 & 0.0176 & 0.0202 & 0.0113 & 0.0106 & 0.0080 & 0.0063                                           & 0.0336 & 0.0292 & 0.0231 & 0.0268 & 0.0100 & 0.0124                            & 0.0040 & 0.0222 & 0.0065 & 0.0363 & 0.0122 & 0.0749 \\
                                                                                                                                                                                                                    
  &  100 & 0.0240 & 0.0287 & 0.0144 & 0.0177 & 0.0105 & 0.0104                                          & 0.0402 & 0.0430 & 0.0258 & 0.0264 & 0.0087 & 0.0094                            & 0.0037 & 0.0225 & 0.0080 & 0.0505 & 0.0135 & 0.0823 \\
                                                                                                                                                                                                                    \midrule
MSE&$v$& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}                               & {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}                     & {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}\\
                                                                                                                                                                                                                 \midrule
$\hat\theta$&    1 & 0.1665 & 0.0398 & 0.1529 & 0.0245 & 0.3150 & 0.0131                                & 1.9196 & 0.0822 & 0.9467 & 0.0518 & 1.3855 & 0.0242                             & 0.0048 & 0.2282 & 0.0028 & 0.1434 & 0.0016 & 0.0746 \\
                                                                                                                                                                                                                    
   & 10 & 0.0119 & 0.0530 & 0.0083 & 0.0350 & 0.0039 & 0.0171                                        & 0.0257 & 0.0998 & 0.0158 & 0.0665 & 0.0077 & 0.0339                              & 0.0039 & 0.2033 & 0.0033 & 0.1548 & 0.0018 & 0.0797 \\
                                                                                                                                                                                                                    
  &  20 & 0.0120 & 0.0530 & 0.0085 & 0.0358 & 0.0043 & 0.0174                                         & 0.0269 & 0.1028 & 0.0167 & 0.0650 & 0.0080 & 0.0346                             & 0.0043 & 0.2106 & 0.0038 & 0.1583 & 0.0018 & 0.0782 \\
                                                                                                                                                                                                                    
 &   100 & 0.0120 & 0.0534 & 0.0076 & 0.0335 & 0.0040 & 0.0177                                        & 0.0235 & 0.1031 & 0.0171 & 0.0646 & 0.0078 & 0.0335                             & 0.0042 & 0.2075 & 0.0033 & 0.1530 & 0.0017 & 0.0783 \\
                                                                                                                                                                                                                    
$\tilde\theta$&    1 & 0.0461 & 0.0395 & 0.0408 & 0.0241 & 0.0515 & 0.0130                                & 0.5264 & 0.0784 & 0.6161 & 0.0502 & 0.5338 & 0.0233                         & 0.0069 & 0.3462 & 0.0081 & 0.3464 & 0.0099 & 0.3710 \\
                                                                                                                                                                                                                    
   & 10 & 0.0117 & 0.0525 & 0.0084 & 0.0350 & 0.0039 & 0.0171                                         & 0.0266 & 0.1002 & 0.0156 & 0.0662 & 0.0077 & 0.0338                         & 0.0084 & 0.3664 & 0.0102 & 0.3831 & 0.0119 & 0.4084 \\
                                                                                                                                                                                                                    
  &  20 & 0.0118 & 0.0525 & 0.0084 & 0.0356 & 0.0042 & 0.0172                                        & 0.0261 & 0.1024 & 0.0166 & 0.0649 & 0.0080 & 0.0346                        & 0.0086 & 0.3675 & 0.0098 & 0.3736 & 0.0116 & 0.4051 \\
                                                                                                                                                                                                                    
 &   100 & 0.0120 & 0.0532 & 0.0076 & 0.0336 & 0.0040 & 0.0177                                        & 0.0229 & 0.1024 & 0.0169 & 0.0643 & 0.0078 & 0.0335                         & 0.0085 & 0.3699 & 0.0098 & 0.3818 & 0.0115 & 0.3975 \\
                                                                                                                                                                                                                    
   $\check\theta$&   1 & 0.0287 & 0.0387 & 0.0301 & 0.0240 & 0.0302 & 0.0128                                & 0.1552 & 0.0769 & 0.1628 & 0.0495 & 0.1677 & 0.0231                         & 0.0016 & 0.1481 & 0.0017 & 0.1136 & 0.0020 & 0.1014 \\
                                                                                                                                                                                                                    
   & 10 & 0.0109 & 0.0512 & 0.0081 & 0.0344 & 0.0038 & 0.0169                                         & 0.0243 & 0.0970 & 0.0150 & 0.0651 & 0.0075 & 0.0335                         & 0.0023 & 0.1610 & 0.0023 & 0.1392 & 0.0027 & 0.1235 \\
                                                                                                                                                                                                                    
  &  20 & 0.0111 & 0.0514 & 0.0081 & 0.0349 & 0.0041 & 0.0170                                         & 0.0242 & 0.0996 & 0.0158 & 0.0639 & 0.0078 & 0.0343                         & 0.0024 & 0.1635 & 0.0024 & 0.1372 & 0.0027 & 0.1279 \\
                                                                                                                                                                                                                    
 &   100 & 0.0115 & 0.0524 & 0.0073 & 0.0331 & 0.0040 & 0.0176                                        & 0.0217 & 0.1008 & 0.0162 & 0.0634 & 0.0076 & 0.0332                        & 0.0022 & 0.1606 & 0.0025 & 0.1415 & 0.0029 & 0.1312 \\
                                                                                                                                                                                                                   \midrule
Size&$v$& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}                              & {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}           & {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}& {$\lambda$}&{$\beta$}\\
                                                                                                                                                                                                                 \midrule 
 $\hat\theta$&    1 & 0.0430 & 0.0575 & 0.0330 & 0.0475 & 0.0480 & 0.0580                               & 0.0540 & 0.0555 & 0.0530 & 0.0565 & 0.0500 & 0.0415                         & 0.0530 & 0.0540 & 0.0380 & 0.0415 & 0.0560 & 0.0535 \\
                                                                                                                                                                                                                    
   & 10 & 0.0610 & 0.0565 & 0.0640 & 0.0470 & 0.0440 & 0.0430                                       & 0.0490 & 0.0445 & 0.0410 & 0.0480 & 0.0420 & 0.0535                         & 0.0520 & 0.0440 & 0.0470 & 0.0580 & 0.0490 & 0.0605 \\
                                                                                                                                                                                                                    
  &  20 & 0.0510 & 0.0550 & 0.0500 & 0.0555 & 0.0660 & 0.0500                                        & 0.0560 & 0.0565 & 0.0500 & 0.0415 & 0.0460 & 0.0535                         & 0.0470 & 0.0520 & 0.0460 & 0.0510 & 0.0430 & 0.0445 \\
                                                                                                                                                                                                                    
 &   100 & 0.0510 & 0.0570 & 0.0450 & 0.0435 & 0.0490 & 0.0520                                       & 0.0470 & 0.0530 & 0.0550 & 0.0505 & 0.0490 & 0.0535                         & 0.0470 & 0.0540 & 0.0450 & 0.0505 & 0.0450 & 0.0455 \\
                                                                                                                                                                                                                    
$\tilde\theta$&    1 & 0.1040 & 0.0585 & 0.0760 & 0.0455 & 0.1070 & 0.0615                               & 0.0640 & 0.0535 & 0.0740 & 0.0610 & 0.0710 & 0.0430                         & 0.8110 & 0.4335 & 0.8890 & 0.6105 & 0.9390 & 0.8100 \\
                                                                                                                                                                                                                    
   & 10 & 0.0560 & 0.0570 & 0.0640 & 0.0480 & 0.0440 & 0.0425                                        & 0.0580 & 0.0485 & 0.0390 & 0.0485 & 0.0490 & 0.0525                         & 0.9070 & 0.4670 & 0.9460 & 0.6540 & 0.9840 & 0.8845 \\
                                                                                                                                                                                                                    
  &  20 & 0.0560 & 0.0540 & 0.0550 & 0.0535 & 0.0590 & 0.0510                                       & 0.0620 & 0.0570 & 0.0500 & 0.0425 & 0.0530 & 0.0555                         & 0.8930 & 0.4820 & 0.9380 & 0.6420 & 0.9760 & 0.8800 \\
                                                                                                                                                                                                                    
 &   100 & 0.0590 & 0.0570 & 0.0490 & 0.0440 & 0.0480 & 0.0520                                       & 0.0440 & 0.0520 & 0.0550 & 0.0505 & 0.0520 & 0.0515                         & 0.8920 & 0.4785 & 0.9430 & 0.6730 & 0.9790 & 0.8735 \\
                                                                                                                                                                                                                    
    $\check\theta$&    1 & 0.0210 & 0.0570 & 0.0150 & 0.0465 & 0.0160 & 0.0585                             & 0.0240 & 0.0545 & 0.0130 & 0.0630 & 0.0120 & 0.0435                & 0.1120 & 0.1040 & 0.1260 & 0.1000 & 0.1510 & 0.1210 \\
                                                                                                                                                                                                                    
   & 10 & 0.0430 & 0.0560 & 0.0600 & 0.0455 & 0.0390 & 0.0445                                        & 0.0520 & 0.0510 & 0.0360 & 0.0510 & 0.0420 & 0.0520                        & 0.1310 & 0.1095 & 0.1560 & 0.1330 & 0.1730 & 0.1475 \\
                                                                                                                                                                                                                    
  &  20 & 0.0510 & 0.0570 & 0.0520 & 0.0545 & 0.0540 & 0.0490                                        & 0.0610 & 0.0575 & 0.0520 & 0.0435 & 0.0470 & 0.0550                         & 0.1440 & 0.1190 & 0.1470 & 0.1245 & 0.1800 & 0.1630 \\
                                                                                                                                                                                                                    
 &   100 & 0.0530 & 0.0575 & 0.0470 & 0.0420 & 0.0490 & 0.0515                                       & 0.0430 & 0.0555 & 0.0580 & 0.0515 & 0.0480 & 0.0520                         & 0.1290 & 0.1165 & 0.1550 & 0.1280 & 0.1730 & 0.1470 \\
                                                                                                                                                                                                                \bottomrule

\end{tabular}
 
\caption{Monte Carlo absolute  bias, mean squared error and size, nominal size 5\%, dependent weight matrices regenerated in each trial.}\label{table_dep_bias_MSE_stoch_t}
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