Microsoft Word - 00_tresc.docx DYNAMIC ECONOMETRIC MODELS Vol. 10 – Nicolaus Copernicus University – Toruń – 2010 Witold Orzeszko Nicolaus Copernicus University in Toruń Measuring Nonlinear Serial Dependencies Using the Mutual Information Coefficient† A b s t r a c t: Construction, estimation and application of the mutual information measure have been presented in this paper. The simulations have been carried out to verify its usefulness to detect nonlinear serial dependencies. Moreover, the mutual information measure has been applied to the indices and the sector sub-indices of the Warsaw Stock Exchange. K e y w o r d s: nonlinearity, mutual information coefficient, mutual information, serial depen- dencies. 1. Introduction Measuring relationships between variables is an extremely important area of research in econometrics. To this end the Pearson correlation coefficient is commonly used. However, the Pearson coefficient is not a proper tool for measuring nonlinear dependencies. Therefore, in the case of nonlinearity other methods must be used. The mutual information coefficient is one of the most important tools to detect nonlinear relationships. It comes from the information theory and is based on a concept of entropy. The mutual information coefficient may be applied to measure dependencies between two time series or serial de- pendencies in a single time series. 2. Measuring Nonlinear Dependencies in Time Series There are various methods to measure nonlinear dependencies in time series (cf. Granger, Terasvirta, 1993; Maasoumi, Racine, 2002; Bruzda, 2004). One of the most important is the mutual information measure (MI hereafter), given by the formula: † Financial support of Nicolaus Copernicus Univerity in Toruń for the project UMK 397-E is gratefully acknowledged. Witold Orzeszko 98 , )()( ),( log),(),( 21 ∫∫ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = dxdy ypxp yxp yxpYXI (1) where ),( yxp is a joint probability density function and )(1 xp and )(2 yp are marginal densities for random variables X and Y. It can be shown that for all X and Y the measure ),( YXI takes non-negative values and 0),( =YXI only if X and Y are independent. It is convenient to define the mutual information coefficient, given by the expression: .1),( ),(2 YXIeYXR −−= (2) It can be shown that the mutual information coefficient has the following properties (cf. Granger, Terasvirta, 1993; Granger, Lin, 1994): 1. 1),(0 ≤≤ YXR , 2. 0),( =YXR ⇔ X and Y are independent, 3. 1),( =YXR ⇔ )( XfY = , where f is some invertible function, 4. R is unaltered if X, Y are replaced by instantaneous transformations )(),( 21 YhXh , i.e. ( ) ( ))(),(, 21 YhXhRYXR = , 5. if ( )YX , (or ( ))(),( 21 YhXh , where 1h and 2h are instantaneous) has a joint Gaussian distribution with correlation ),( YXρ , then ),(),( YXYXR ρ= . In the literature one can find several methods for estimating a value of ),( YXI . Essentially, due to the technique of estimating the probability density functions in Equation 1, they can be divided into three main groups (cf. Dioni- sio, Menezes, Mendes, 2003): − histogram-based estimators, − kernel-based estimators, − parametric methods. The kernel-based estimators have many adjustable parameters such as the optimal kernel width and the optimal kernel form, and a non-optimal choice of those parameters may cause a large bias in the results. For the application of parametric methods one needs to know the specific form of the generating process (Dionisio, Menezes, Mendes, 2003)). Therefore a standard way is to estimate the densities by means of histograms (cf. Darbellay, Wuertz, 2000). One can also define auto mutual information at lag k for a stationary dis- crete-valued stochastic process nXXX ,...,, 21 as the mutual information be- tween random variables tX and ktX + : Measuring Nonlinear Serial Dependencies Using the Mutual Information Coefficient 99 . )()( ),( log),(),( ∑∑ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = + + ++ t ktx x ktt ktt kttktt xPxP xxP xxPXXI (3) Since the process is stationary, ),( ktt XXI + is independent of t and so we can refer to the mutual information at lag k, as )(kI (Fonseca, Crovella, Sala- matian, 2008). This means that, the mutual information measure may be used to measure serial dependencies in a single time series as well. To this end, the past realiza- tions of the investigated data X should be taken as the variable Y. It should be emphasized that MI measures both linear and nonlinear depen- dencies, so to identify serial nonlinear relationships, analyzed data must be pre- filtered by an estimated ARMA-type model. 3. Application of the Mutual Information Measure to Detect Serial Dependencies 3.1. Simulated Data The aim of the simulations was to verify, if the mutual information measure may be effectively applied to detect nonlinear serial dependencies. The time series produced from five different generating models and two different sample sizes (with each of those models) were used in the simulations. This data was generated by Barnett et al. (1998) to compare the power of some popular tests for nonlinearity and chaos1. Specifically, these were: five time series of 2000 observations – M1, M2, M3, M4, M5 and five time series of their first 380 observations – M1s, M2s, M3s, M4s, M5s. The investigated series were generated from the following models2: I) M1 – logistic map3: ),1(57.3 11 −− −= ttt xxx (4) II) M2 – GARCH(1,1) process: ,ttt uhx = (5a) ,8.01.01 1 2 1 −− ++= ttt hxh (5b) where 10 =h and 00 =x . 1 The data was downloaded from the homepage of W.A. Barnett: http://econ.tepper.cmu.edu/ /barnett/Papers.html. 2 In all cases, the white-noise disturbances – ut were sampled independently from the standard normal distribution. 3 The logistic map with the parameter equaled to 3.57 generates chaotic dynamics. Witold Orzeszko 100 III) M3 – Nonlinear Moving Average Process (NLMA): ,8.0 21 −−+= tttt uuux (6) IV) M4 – ARCH(1) process: ,5.01 2 1 ttt uxx −+= (7) V) M5 – ARMA(2,1) process: ,3.015.08.0 121 −−− +++= ttttt uuxxx (8) where 10 =x and 7.01 =x . In each case the mutual information measure was calculated for the raw series and for its residuals from the fitted ARMA model. First, stationarity was verified using the Augmented Dickey-Fuller test. The null hypothesis of a unit root was strongly rejected for the all investigated data, except M5s. Thus, instead of M5s, the series of its first differences – M5s_diff was chosen for further research. In Table 1 the ARMA models fitted to analyzed series are presented4. Table 1. ARMA models for the generated series Series ARMA model Series ARMA model M1 White noise (EX=0.648) M1s White noise (EX=0.649) M2 White noise (EX=0.034) M2s White noise (EX=0.067) M3 White noise (EX= 0.007) M3s White noise (EX= 0.033) M4 White noise (EX= 0.011) M4s White noise (EX= 0.018) M5 ARMA(1,1) M5s_diff MA(1) Next, the Ljung-Box test was applied to test if the residual series are white noise. The test confirmed that no investigated residuals contain linear depen- dencies. To estimate the mutual information measure the method proposed by Fraser and Swinney (1986) was used5. This method is based on an analysis of the two- dimensional histogram. Briefly speaking, it consists in covering the two- dimensional plane containing pairs ( )tt yx , with rectangular partitions and cal- culating frequencies of points in each partition. Next, Equation 1 is used, i.e. the calculated frequencies are estimators of the probability density functions and the integration is carried out numerically. Let ki denotes an estimated value of the mutual information measure be- tween variables tX and ktX − . Due to a purpose of the research, the key task is to verify the hypothesis of mutual information measure’s insignificance (i.e the hypothesis of independence). To this end, for each investigated series and for 4 The models were selected based on the Schwarz criterion. 5 In the calculations the m-file created by A. Leontitsis was used. Measuring Nonlinear Serial Dependencies Using the Mutual Information Coefficient 101 each 10...,,2,1=k , the p-value was evaluated through bootstraping6 with 00010 repetitions7. In Tables 2-6 the calculated values of ki and the corres- ponding p-values (at the bottom) are summarized. The p-values not larger than 0.005 are bolded8. Table 2. Values of ki for M1s and M1 k series 1 2 3 4 5 6 7 8 9 10 M1s 1.6927 0.0000 1.6963 0.0000 1.6123 0.0000 1.7148 0.0000 1.5919 0.0000 1.6849 0.0000 1.5412 0.0000 1.6381 0.0000 1.5379 0.0000 1.6560 0.0000 M1 2.0139 0.0000 2.0090 0.0000 2.0064 0.0000 2.2520 0.0000 1.9981 0.0000 1.9991 0.0000 1.9940 0.0000 2.2737 0.0000 1.9891 0.0000 1.9891 0.0000 Table 3. Values of ki for M2s and M2 k series 1 2 3 4 5 6 7 8 9 10 M2s 0.0848 0.9616 0.1538 0.0201 0.1191 0.3802 0.1308 0.1786 0.1231 0.3052 0.1616 0.0081 0.1701 0.0029 0.1162 0.4412 0.1281 0.2187 0.1228 0.3090 M2 0.0541 0.0053 0.0562 0.0025 0.0477 0.0808 0.0488 0.0536 0.0492 0.0451 0.0509 0.0227 0.0541 0.0052 0.0461 0.1303 0.0449 0.1868 0.0334 0.9315 Table 4. Values of ki for M3s and M3 k series 1 2 3 4 5 6 7 8 9 10 M3s 0.1857 0.0492 0.1586 0.3316 0.1425 0.6241 0.1469 0.5429 0.1323 0.8032 0.1028 0.9927 0.1897 0.0353 0.1600 0.3096 0.1525 0.4389 0.1606 0.2987 M3 0.0725 0.0000 0.0658 0.0001 0.0307 0.9634 0.0429 0.2065 0.0309 0.9599 0.0383 0.5426 0.0372 0.6274 0.0404 0.3724 0.0389 0.4868 0.0456 0.0976 Table 5. Values of ki for M4s and M4 k series 1 2 3 4 5 6 7 8 9 10 M4s 0.1365 0.2663 0.1667 0.0205 0.1442 0.1562 0.1349 0.2940 0.1198 0.6104 0.1367 0.2613 0.1347 0.2959 0.1327 0.3361 0.1435 0.1641 0.1464 0.1303 M4 0.1053 0.0000 0.0472 0.0051 0.0363 0.3383 0.0379 0.2324 0.0286 0.9261 0.0344 0.5058 0.0370 0.2866 0.0475 0.0039 0.0368 0.3074 0.0344 0.5059 6 Bootstrap without replacement (i.e. permutation) was performed. Bootstrapped p-values cor- respond to a one-sided test. 7 In this way, for each of the filtered series an expected distribution of MI(1) (i.e. the MI measure with k=1) was determined. Next, this distribution has led to evaluation of the p-value for each k=1,2,...,10. 8 Note that the rejection of the null of ki insignificance for at least one k=1,2,...,10 implies the rejection of the hypothesis of serial independence. Therefore, adopting the value 0.005 for each k implies that the probability for a type I error (in the test of serial independence) is approximately 5%. Witold Orzeszko 102 Table 6. Values of ki for M5s and M5 k series 1 2 3 4 5 6 7 8 9 10 M5s 1.4787 0.0000 1.1206 0.0000 0.9817 0.0000 0.8640 0.0000 0.7505 0.0000 0.6895 0.0000 0.6344 0.0000 0.6310 0.0000 0.6173 0.0000 0.6070 0.0000 M5s_ diff 0.1390 0.5519 0.1658 0.1199 0.1288 0.7509 0.1438 0.4542 0.1496 0.3452 0.2012 0.0039 0.1642 0.1351 0.1297 0.7340 0.1161 0.9125 0.1387 0.5560 M5s_ diffMA 0.1224 0.7971 0.1584 0.1595 0.1225 0.7942 0.1242 0.7668 0.1444 0.3745 0.1391 0.4816 0.1624 0.1193 0.1510 0.2584 0.1495 0.2821 0.1474 0.3179 M5 1.7145 0.0000 1.3154 0.0000 1.0949 0.0000 0.9504 0.0000 0.8414 0.0000 0.7597 0.0000 0.6958 0.0000 0.6449 0.0000 0.5917 0.0000 0.5584 0.0000 M5ARMA 0.0422 0.2714 0.0375 0.6530 0.0417 0.3103 0.0412 0.3438 0.0355 0.8012 0.0396 0.4685 0.0419 0.2963 0.0486 0.0398 0.0434 0.2030 0.0397 0.4640 In Tables 7-8 the results of nonlinearity detection carried out by the MI measure are summarized. Table 7. Results of nonlinearity detection for the long series Series Serial dependencies Nonlinearity M1 YES YES M2 YES YES M3 YES YES M4 YES YES M5 YES NO Table 8. Results of nonlinearity detection for the short series Series Serial dependencies Nonlinearity M1s YES YES M2s YES YES M3s NO NO M4s NO NO M5s_ diff YES NO As it is clearly seen, the MI measure correctly identified each of the investi- gated long series. In an application to the short series it led to erroneous conclu- sions in the case of M3s and M4s. The obtained result is consistent with studies by other authors, i.e. it indicates that histogram-based estimators may be unreli- able in a case of a small number of observations (e.g. Dionisio, Menezes, Men- des, 2003). 3.2. Stock Market Indices In this section the indices and the sector sub-indices of the Warsaw Stock Exchange from 2.01.2001–15.04.2009 (2078 observations) were analyzed. For the each index, the three time series were investigated: daily log returns, residu- als from their ARMA and ARMA-GARCH models. Investigation of the residu- als from the ARMA model gives information, if dependencies are nonlinear. If so, the standardized residuals from the ARMA-GARCH model were ana- Measuring Nonlinear Serial Dependencies Using the Mutual Information Coefficient 103 lyzed, to verify if this class of processes can capture nonlinear dynamics found in the investigated data9. The results of this analysis are presented in Tables 9-20. Table 9. Values of ki for the WIG index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0458 0.0000 0.0444 0.0003 0.0605 0.0000 0.0612 0.0000 0.0486 0.0000 0.0518 0.0000 0.0350 0.0338 0.0365 0.0153 0.0522 0.0000 0.0559 0.0000 MA(1) 0.0412 0.0010 0.0455 0.0000 0.0549 0.0000 0.0632 0.0000 0.0427 0.0002 0.0500 0.0000 0.0379 0.0074 0.0313 0.1530 0.0552 0.0000 0.0566 0.0000 MA(1)- GARCH(3,1) 0.0458 0.0225 0.0498 0.0033 0.0336 0.7074 0.0395 0.2254 0.0359 0.5142 0.0306 0.8964 0.0302 0.9124 0.0352 0.5738 0.0328 0.7700 0.0309 0.8823 Table 10. Values of ki for the WIG20 index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0514 0.0000 0.0415 0.0106 0.0577 0.0000 0.0690 0.0000 0.0489 0.0002 0.0509 0.0000 0.0388 0.0381 0.0388 0.0373 0.0438 0.0029 0.0537 0.0000 MA(1) 0.0456 0.0011 0.0471 0.0006 0.0579 0.0000 0.0687 0.0000 0.0506 0.0001 0.0510 0.0001 0.0402 0.0187 0.0458 0.0009 0.0439 0.0028 0.0545 0.0000 MA(1)- GARCH(3,1) 0.0441 0.0410 0.0457 0.0222 0.0337 0.6723 0.0382 0.2962 0.0311 0.8499 0.0333 0.6978 0.0307 0.8683 0.0384 0.2806 0.0303 0.8867 0.0272 0.9756 Table 11. Values of ki for the mWIG40 index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0728 0.0000 0.0508 0.0000 0.0630 0.0000 0.0603 0.0000 0.0545 0.0000 0.0660 0.0000 0.0508 0.0000 0.0343 0.0058 0.0397 0.0002 0.0428 0.0000 AR(3) 0.0511 0.0000 0.0458 0.0000 0.0539 0.0000 0.0569 0.0000 0.0508 0.0000 0.0462 0.0000 0.0465 0.0000 0.0376 0.0002 0.0379 0.0002 0.0460 0.0000 AR(3)- GARCH(1,2) 0.0340 0.0964 0.0301 0.3434 0.0278 0.5657 0.0404 0.0039 0.0264 0.6980 0.0377 0.0182 0.0250 0.8188 0.0309 0.2750 0.0283 0.5131 0.0295 0.3955 Table 12. Values of ki for the sWIG80 index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0911 0.0000 0.0551 0.0000 0.0680 0.0000 0.0579 0.0000 0.0597 0.0000 0.0546 0.0000 0.0498 0.0000 0.0416 0.0014 0.0440 0.0003 0.0426 0.0006 ARMA(1.2) 0.0478 0.0000 0.0386 0.0031 0.0538 0.0000 0.0502 0.0000 0.0397 0.0020 0.0479 0.0000 0.0371 0.0074 0.0340 0.0349 0.0369 0.0083 0.0376 0.0056 ARMA(1,2)- GARCH(1,1) 0.0268 0.7878 0.0300 0.5014 0.0367 0.0616 0.0282 0.6646 0.0258 0.8545 0.0345 0.1451 0.0255 0.8722 0.0295 0.5471 0.0278 0.7014 0.0309 0.4072 9 The fit of all estimated models was positively verified using the Box-Ljung and the Engle tests. Table 13. Values of ki for the WIG-Banking index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0429 0.0000 0.0439 0.0000 0.0628 0.0000 0.0556 0.0000 0.0485 0.0000 0.0518 0.0000 0.0476 0.0000 0.0516 0.0000 0.0602 0.0000 0.0443 0.0000 MA(1) 0.0421 0.0000 0.0469 0.0000 0.0566 0.0000 0.0542 0.0000 0.0609 0.0000 0.0530 0.0000 0.0421 0.0000 0.0544 0.0000 0.0565 0.0000 0.0496 0.0000 MA(1)- GARCH(1,2) 0.0387 0.0534 0.0346 0.2279 0.0347 0.2242 0.0278 0.8131 0.0308 0.5525 0.0320 0.4328 0.0276 0.8250 0.0302 0.6136 0.0306 0.5702 0.0354 0.1790 Table 14. Values of ki for the WIG-Construction index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0525 0.0000 0.0301 0.2070 0.0415 0.0009 0.0400 0.0016 0.0365 0.0119 0.0451 0.0001 0.0460 0.0001 0.0326 0.0823 0.0321 0.1004 0.0451 0.0001 ARMA(2,1) 0.0336 0.0145 0.0386 0.0003 0.0428 0.0000 0.0387 0.0003 0.0350 0.0064 0.0365 0.0022 0.0391 0.0002 0.0221 0.7270 0.0320 0.0311 0.0481 0.0000 ARMA(2,1)- GARCH(1,1) 0.0286 0.5966 0.0289 0.5661 0.0321 0.2637 0.0293 0.5281 0.0239 0.9422 0.0305 0.4099 0.0263 0.8061 0.0251 0.8875 0.0301 0.4452 0.0338 0.1569 Table 15. Values of ki for the WIG-Developers index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.1392 0.0063 0.1477 0.0013 0.1290 0.0246 0.1154 0.1188 0.1859 0.0000 0.1292 0.0240 0.1255 0.0392 0.1370 0.0091 0.1699 0.0000 0.1353 0.0118 ARMA(1,1) 0.1479 0.0022 0.1562 0.0006 0.1466 0.0028 0.1144 0.1664 0.1506 0.0017 0.1515 0.0014 0.1488 0.0021 0.1258 0.0531 0.1412 0.0079 0.1484 0.0022 ARMA(1,1)- GARCH(1,2) 0.0928 0.9250 0.1147 0.5124 0.0999 0.8324 0.0929 0.9245 0.1195 0.3976 0.1153 0.4980 0.1189 0.4135 0.1199 0.3861 0.0869 0.9708 0.1241 0.3014 Table 16. Values of ki for the WIG-Food index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0544 0.0000 0.0361 0.0000 0.0449 0.0000 0.0473 0.0000 0.0446 0.0000 0.0311 0.0037 0.0314 0.0028 0.0337 0.0003 0.0305 0.0052 0.0342 0.0002 ARMA(1,1) 0.0371 0.0000 0.0365 0.0000 0.0418 0.0000 0.0433 0.0000 0.0358 0.0000 0.0270 0.0055 0.0310 0.0003 0.0409 0.0000 0.0230 0.0601 0.0298 0.0007 ARMA(1,1)- GARCH(1,1) 0.0311 0.5233 0.0340 0.2766 0.0338 0.2873 0.0347 0.2198 0.0239 0.9738 0.0309 0.5448 0.0281 0.7944 0.0281 0.7938 0.0311 0.5252 0.0362 0.1404 Measuring Nonlinear Serial Dependencies Using the Mutual Information Coefficient 105 Table 17. Values of ki for the WIG-IT index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0486 0.0000 0.0388 0.0038 0.0466 0.0000 0.0573 0.0000 0.0454 0.0000 0.0443 0.0001 0.0335 0.0580 0.0513 0.0000 0.0469 0.0000 0.0434 0.0002 AR(1) 0.0585 0.0000 0.0359 0.0449 0.0476 0.0000 0.0619 0.0000 0.0553 0.0000 0.0488 0.0000 0.0314 0.2556 0.0499 0.0000 0.0543 0.0000 0.0409 0.0032 AR(1)- GARCH(1,1) 0.0362 0.0778 0.0251 0.8876 0.0270 0.7622 0.0339 0.1810 0.0222 0.9796 0.0260 0.8343 0.0282 0.6611 0.0244 0.9176 0.0303 0.4641 0.0293 0.5646 Table 18. Values of ki for the WIG-Media index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0481 0.0555 0.0560 0.0049 0.0448 0.1229 0.0562 0.0047 0.0475 0.0642 0.0456 0.1020 0.0350 0.6139 0.0422 0.2144 0.0304 0.8539 0.0393 0.3531 MA(1) 0.0484 0.1063 0.0571 0.0097 0.0516 0.0473 0.0529 0.0333 0.0519 0.0432 0.0398 0.4922 0.0464 0.1644 0.0450 0.2159 0.0446 0.2322 0.0426 0.3298 MA(1)- GARCH(1,1) 0.0484 0.1380 0.0414 0.4673 0.0427 0.3925 0.0481 0.1451 0.0510 0.0735 0.0352 0.8260 0.0397 0.5785 0.0363 0.7691 0.0465 0.2020 0.0370 0.7320 Table 19. Values of ki for the WIG-Oil&Gas index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0825 0.0185 0.0780 0.0503 0.0761 0.0733 0.0711 0.1660 0.0862 0.0076 0.0658 0.3335 0.0562 0.7510 0.0685 0.2422 0.0829 0.0166 0.0667 0.3009 AR(2) 0.0816 0.0203 0.0652 0.3279 0.0619 0.4671 0.0824 0.0183 0.0820 0.0195 0.0863 0.0062 0.0573 0.6711 0.0823 0.0184 0.0878 0.0043 0.0771 0.0493 AR(2)- GARCH(1,1) 0.0451 0.8837 0.0611 0.2362 0.0493 0.7406 0.0573 0.3809 0.0716 0.0368 0.0573 0.3787 0.0524 0.6043 0.0652 0.1240 0.0592 0.3055 0.0472 0.8178 Table 20. Values of ki for the WIG-Telecom index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0467 0.0072 0.0395 0.1307 0.0429 0.0369 0.0687 0.0000 0.0440 0.0234 0.0393 0.1405 0.0417 0.0579 0.0469 0.0062 0.0419 0.0518 0.0514 0.0007 GARCH(1.3) 0.0311 0.5186 0.0340 0.2752 0.0338 0.2834 0.0347 0.2234 0.0239 0.9693 0.0309 0.5411 0.0281 0.7826 0.0281 0.7821 0.0311 0.5207 0.0362 0.1448 The results summarized in Tables 9-20 indicate that evidence of serial de- pendencies was found for the most investigated indices10. The same conclusion may be drawn for the residuals from the ARMA models, which means that the detected dependencies are nonlinear. In most cases the estimated ARMA- GARCH models were able to capture these nonlinearities. Only in the case 10 The exception is the WIG-Oil&Gas index. In this case the obtained result is rather unusual, i.e. filtering data by the ARMA model caused the appearance of significance of the MI measure. 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Współczynnik informacji wzajemnej jako miara zależności nieliniowych w szeregach czasowych Z a r y s t r e ś c i. W artykule scharakteryzowano konstrukcję, estymację oraz możliwości zasto- sowania współczynnika informacji wzajemnej. Przedstawiono wyniki symulacji, prowadzących do weryfikacji jego przydatności w procesie identyfikacji zależności nieliniowych w szeregach czasowych. Ponadto zaprezentowano wyniki zastosowania tego współczynnika do analizy indek- sów Giełdy Papierów Wartościowych w Warszawie. S ł o w a k l u c z o w e: nieliniowość, współczynnik informacji wzajemnej, mutual information, identyfikacja zależności.