Springer Texts in Business and Economics

Autoregressive Conditional Heteroskedasticity

Financial time-series such as foreign exchange rates, inflation rates and stock prices may exhibit some volatility which varies over time. In the case of inflation or foreign exchange rates this could be due to changes in the Federal Reserve’s policies. In the case of stock prices this could be due to rumors about a certain company’s merger or takeover. This suggests that the variance of these time-series may be heteroskedastic. Engle (1982) modeled this heteroskedasticity by relating the conditional variance of the disturbance term at time t to the size of the squared disturbance terms in the recent past. A simple Autoregressive Conditionally Heteroskedastic
(ARCH) model is given by

= E(ut/( t) = Yo + Y iut-i + •• + Yp^t-v (14-22)

where Zt denotes the information set upon which the variance of ut is to be conditioned. This typically includes all the information available prior to period t. In (14.22), the variance of ut conditional on the information prior to period t is an autoregressive function of order p in squared lagged values of ut. This is called an ARCH(p) process. Since (14.22) is a variance, this means that all the Y^s for i = 0,1,-,p have to be non-negative. Engle (1982) showed that a simple test for homoskedasticity, i. e., Ho; yi = Y2 = •• = Yv = 0, can be based upon an ordinary F-test which regresses the squared OLS residuals (e2) on their lagged values (e^^-^e2^) and a constant. The F-statistic tests the joint significance of the regressors and is reported by most regression packages. Alternatively, one can compute T times the centered R2 of this regression and this is distributed as xp under the null hypothesis Ho. This test resembles the usual homoskedasticity tests studied in Chapter 5 except that the squared OLS residuals are regressed upon their lagged values rather than some explanatory variables.

The simple ARCH(1) process

at = Yo + Yiut-1 (14.23)

can be generated as follows: ut = [yo + Y 1ut-i1/‘2€t where et ~ IID(0,1). Note that the sim­plifying variance of unity for et can be achieved by rescaling the parameters yo and y1. In this case, the conditional mean of ut is given by

E(ut/Z t) = [Y o + Y iu2-i}1/2 E (Y/Z t) = 0

since ut 1 is known at time t. Similarly, the conditional variance can be easily obtained from

E(u2/Z t) = [Y o + Y 1u2-1}E (H2/C t) = Yo + Y 1u2-1

since E(e‘2) = 1. Also, the conditional covariances can be easily shown to be zero since

E(utut-s/Z t )= ut-s E(ut/Z t) = 0 for s = 1, 2,^,t

The unconditional mean can be obtained by taking repeated conditional expectations period by period until we reach the initial period, see the Appendix to Chapter 2. For example, taking the conditional expectation of E(ut/Zt) based on information prior to period t — 1, we get

E[E(ut/Zt)/Zt-1 = E (0/Zt-1) = 0

It is clear that all prior conditional expectations of zero will be zero so that E(ut) = 0. Similarly, taking the conditional expectations of E(u2/Zt) based on information prior to period t — 1, we

get

By taking repeated conditional expectations one period at a time we finally get E(ut) = Yo(1 + Y1 + Y1 + •• + Y 1-1) + Y1uo

As t ^ x, the unconditional variance of ut is given by a2 = var(ut) = 70/(1 — Y1) for |y11 < 1 and Yo > 0. Therefore, the ARCH(1) process is homoskedastic.

ARCH models can be estimated using feasible GLS or maximum likelihood methods. Alterna­tively, one can use a double-length regression procedure suggested by Davidson and MacKinnon (1993) to obtain (i) one-step efficient estimates starting from OLS estimates or (ii) the max­imum likelihood estimates. Here we focus on the feasible GLS procedure suggested by Engle (1982). For the regression model

y = Хв + u (14.25)

where y is T x 1 and X is T x k. First, obtain the OLS estimates вOLS and the OLS residuals e. Second, perform the following regression: ef = ao + a+e‘2-1 + residuals. This yields a test for homoskedasticity. Third, compute af = ao + a1 e2-1 and regress [(e2/at) — 1] on (1/at) and

(ef-1 /at). Call the regression estimates da. One updates a' = (ao, a1) by computing a = a + da. Fourth, recompute?2 using the updated a from step 3, and form the set of regressors xtjrt for j = 1,...,k, where

Finally, regress (etst/rt) where

st = 1 — 71 (X —

at at+1 at+1

on xtjrt for j = 1,...,k and obtain the least squares coefficients dy. Update the estimate of в by computing [3 = eOLS + dp. This procedure can run into problems if the are not all positive, see Judge et al. (1985) and Engle (1982) for details.

The ARCH model has been generalized by Bollerslev (1986). The Generalized ARCH (GARCH (p, q)) model can be written as

a2 = Yo + E i=1 Yiu-i + E q=1 Sj a2t-j

In this case, the conditional variance of ut depends upon q of its lagged values as well as p squared lagged values of ut. The simple GARCH (1,1) model is given by

2 і 2 і c 2

at = Yo + Y1ut-1 + S1at-1

An LM test for GARCH (p, q) turns out to be equivalent to testing ARCH (p + q). This simply regresses squared OLS residuals on (p + q) of its squared lagged values. The test statistic is T times the uncentered R2 and is asymptotically distributed as Xp+q under the null of homoskedasticity.

In conclusion, a lot of basic concepts have been introduced in this chapter and we barely scratched the surface. Hopefully, this will motivate the reader to take the next econometrics time series course.

Problems

1. For the AR(1) model

yt = PVt-1 + et t = 1, 2,...,T; with p < 1 and et ~ IIN(0,ct^ )

(a) Show that if yo ~ N(0,a2e/1 — p2), then E(yt) = 0 for all t and var(yt) = oj(1 — p2) so that the mean and variance are independent of t. Note that if p =1 then var(yt) is to. If p > 1 then var(yt) is negative!

(b) Show that cov(yt, yt-s) = psa2 which is only dependent on s, the distance between the two time periods. Conclude from parts (a) and (b) that this AR(1) model is weakly stationary.

(c) Generate the above AR(1) series for T = 250, a2 = 0.25 and various values of p = ±0.9, ±0.8, ±0.5, ±0.3 and ±0.1. Plot the AR(1) series and the autocorrelation function ps versus s.

2. For the MA(1) model

yt = et + Oe—1 t = 1, 2,...,T; with et ~ IIN(0, a2e)

(a) Show that E(yt) = 0 and var(yt) = 1 + в2) so that the mean and variance are independent of t.

(b) Show that cov(yt, yt-1) = ва2 and cov(yt, yt-s) = 0 for s > 1 which is only dependent on s, the distance between the two time periods. Conclude from parts (a) and (b) that this MA(1) model is weakly stationary.

(c) Generate the above MA(1) series for T = 250, a2 = 0.25 and various values of в = 0.9, 0.8, 0.5, 0.3 and 0.1. Plot the MA(1) series and the autocorrelation function versus s.

3. Using the consumption-personal disposable income data for the U. S. used in this chapter:

(a) Compute the sample autocorrelation function for personal disposable income (Yt). Plot the sample correlogram. Repeat for the first-differenced series (AYt). Compute the Ljung-Box Qlb statistic, test that Ho; ps = 0 for s = 1,..., 20.

(b) Run the Augmented Dickey-Fuller test for the existence of a unit root in personal disposable income (Yt).

(c) Define Yt = AYt and run AYt on Yt-1 and a constant and trend. Test that the first-differenced series of personal disposable income is stationary. What do you conclude? Is Yt an I(1) process?

(d) Replicate the regression in (14.21) and verify the Engle-Granger (1987) test for cointegration.

(e) Replicate the GARCH(1,1) model given in Table 14.3.

(f) Repeat parts (a) through (e) using logC and logY. Are there any changes in the above results?

4. (a) Generate T = 25 observations on xt and yt as independent random walks with IIN(0,1)

disturbances. Run the regression yt = a + /3xt + ut and test the null hypothesis Ho; в = 0 using the usual t-statistic at the 1%, 5% and 10% levels. Repeat this experiment 1000 times and report the frequency of rejections at each significance level. What do you conclude?

(b) Repeat part (a) for T = 100 and T = 500.

(c) Repeat parts (a) and (b) generating xt and yt as independent random walks with drift as described in (14.11), using IIN(0,1) disturbances. Let 7 = 0.2 for both series.

(d) Repeat parts (a) and (b) generating xt and yt as independent trend stationary series as described in (14.10), using IIN(0,1) disturbances. Let a =1 and в = 0.04 for both series.

(e) Report the frequency distributions of the R2 statistics obtained in parts (a) through (d) for each sample size and method of generating the time-series. What do you conclude? Hint: See the Monte Carlo experiments in Granger and Newbold (1974), Davidson and MacKinnon (1993) and Banerjee, Dolado, Galbraith and Hendry (1993).

5. For the Money Supply, GNP and interest rate series data for the U. S. given on the Springer web site as MACRO. ASC, fit a VAR three equation model using:

(a) Two lags on each variable.

(b) Three lags on each variable.

(c) Compute the Likelihood Ratio test for part (a) versus part (b).

(d) For the two-equation VAR of Money Supply and interest rate with three lags on each variable, test that the interest rate does not Granger cause the money supply?

(e) How sensitive are the tests in part (d) if we had used only two lags on each variable.

6. For the simple Deterministic Time Trend Model Vt = a + fit + Ut t = 1, ..,T

where ut ~ IIN(0,v2).

(a) Show that

aOLS — a

Pols — P

where the t-th observation of X, the matrix of regressors, is [1,t].

(b) Use the results that ^t=i t = T(T + 1)/2 and t=i t2 = T(T + 1)(2T + 1)/6 to show that

plim (X'X/T) as T is not a positive definite matrix.

(c) Use the fact that

A(X 'X )-1AA-1 (X'u) = (A-1(X 'X )A-1)-1A-1(X'u)

(d) Show that z1 = f=1 ut/VT is N(0,v2) and z2

1)/6T2) with cov(z1,z2) = (T + 1)<t2/2T, so that

z1

z2

(e) Using the results in parts (c) and (d), conclude that the asymptotic distribution of

rVT^a°LS ^ is N(0,a2Q-1). Since (3OLS has the factor TVT rather than the usual

T T(PoLS - P) )

VT, it is said to be superconsistent. This means that not only does (POLS — P) converge to zero in probability limits, but so does T(POLS — P). Note that the normality assumption is not needed for this result. Using the central limit theorem, all that is needed is that ut is White noise with finite fourth moments, see Sims, Stock and Watson (1990) or Hamilton (1994).

7. Test of Hypothesis with a Deterministic Time Trend Model. This is based on Hamilton (1994). In problem 6, we showed that aOLS and POLS converged at different rates, %/T and TVT respectively. Despite this fact, the usual least squares t and F-statistics are asymptotically valid even when the ut’s are not Normally distributed.

(a) Show that s2 = ^'T=1(yt — aOLS — Polst)2/(T — 2) has plim s2 = a2.

(b) In order to test Ho; a = ao, the usual least squares package computes

ta = (Sols — ao)/[s2(1,0)(X'X)-1(1, 0)']1/2

where (X'X) is given in problem 6. Multiply the numerator and denominator by %/T and use the results of part (c) of problem 6 to show that this t-statistic has the same asymptotic distribution as t*a = %/T(aOLS — ao)/aZqu where q11 is the (1, 1) element of Q-1 defined in problem 6. t*a has an asymptotic N(0,1) distribution using the results of part (e) in problem 6.

(c) Similarly, to test Ho; в = во, the usual least squares package computes

te = (Pols — e)/[s2(0,1)(X 'X )-1 (0,1)']1/2.

Multiply the numerator and denominator by T/T and use the results of part (c) of problem 6 to show that this t-statistic has the same asymptotic distribution as t*p = T/T(POLS — P)/a/q22 where q22 is the (2, 2) element of Q-1 defined in problem 6. t*g has an asymptotic N(0, 1) distribution using the results of part (e) in problem 6.

8. A Random Walk Model. This is based on Fuller (1976) and Hamilton (1994). Consider the following random walk model

yt = yt-1 + ut t = 0,1,...,T where ut ~ IIN(0, a2) and yo = 0.

(a) Show that yt can be written as yt = u1 + u2 + .. + ut with E(yt) = 0 and var(yt) = ta2 so that yt ~ N(0,ta2).

(b) Square the random walk equation yj = (yt-1 + ut)2 and solve for yt-1ut. Sum this over t = 1, 2,.. .,T and show that

St=1 yt-1ut = (yT/2) — ^21=1 ut/2

Divide by Ta2 and show that ^T=1 yt-1ut/Ta2 is asymptotically distributed as (y2 — 1)/2. Hint: Use the fact that yT ~ N(0,Ta2).

(c) Using the fact that yt-1 ~ N(0, (t — 1)a2) show that E (Y^t=1 y‘t-1j = a2T(T — 1)/2. Hint: Use the expression for ^T=11 in problem 6.

(d) Suppose we had estimated an AR(1) model rather than a random walk, i. e., yt = pyt-1 + ut when the true p = 1. The OLS estimate is

X^T X^T 2 і X^T X^T 2

p = t=1 yt-1 rytl t=1 yt-1 = p + t=1 yt-1 ut/ t=1 yt-1

Show that

t=1 yt-1ut/Ta

plim T(p — p) = plim <tTL 2 2 2 = 0

t=1 y - 1/T 2 a2

Note that the numerator was considered in part (b), while the denominator was considered in part (c). One can see that the asymptotic distribution of p when p = 1 is a ratio of (x2 — 1)/2 random variable to a non-standard distribution in the denominator which is beyond the scope of this book, see Hamilton (1994) or Fuller (1976) for further details. The object of this exercise is to show that if p = 1, VT(s — p) is no longer normal as in the standard stationary least squares regression with p < 1. Also, to show that for the nonstationary (random walk) model, p converges at a faster rate (T) than for the stationary case (%/T). From part (c) it is clear that one has to divide the denominator of p by T2 rather than T to get a convergent distribution.

9. Consider the cointegration example given in (14.13) and (14.14).

(a) Verify equations (14.15)-(14.20).

(b) Show that the OLS estimator of в obtained by regressing Ct on Yt is superconsistent, i. e., show that plim T(Pols — в) ^ 0 as T ^x>.

References

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0 z

Ф(1.65) = pr[z < 1.65] = 0.9505

Table A Area under the Standard Normal Distribution

B. H. Baltagi, Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-20059-5, © Springer-Verlag Berlin Heidelberg 2011

Pr[t8 >ta = 2.306] = 0.025

Table B Right-Tail Critical Values for the t-Distribution

V-2/v 1 2 3 4 5 6 7 8 9 10 12 15 20 25 30 40 1 161.448 199.500 215.707 224.583 230.162 233.986 236.768 238.883 240.543 241.882 243.906 245.950 248.013 249.260 250.095 251.143 2 18.513 19.000 19.164 19.247 19.296 19.330 19.353 19.371 19.385 19.396 19.413 19.429 19.446 19.456 19.462 19.471 3 10.128 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786 8.745 8.703 8.660 8.634 8.617 8.594 4 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.999 5.964 5.912 5.858 5.803 5.769 5.746 5.717 5 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.772 4.735 4.678 4.619 4.558 4.521 4.496 4.464 6 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.099 4.060 4.000 3.938 3.874 3.835 3.808 3.774 7 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.677 3.637 3.575 3.511 3.445 3.404 3.376 3.340 8 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.388 3.347 3.284 3.218 3.150 3.108 3.079 3.043 9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.179 3.137 3.073 3.006 2.936 2.893 2.864 2.826 10 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020 2.978 2.913 2.845 2.774 2.730 2.700 2.661 11 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.896 2.854 2.788 2.719 2.646 2.601 2.570 2.531 12 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.796 2.753 2.687 2.617 2.544 2.498 2.466 2.426 13 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.714 2.671 2.604 2.533 2.459 2.412 2.380 2.339 14 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.646 2.602 2.534 2.463 2.388 2.341 2.308 2.266 15 4.543 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588 2.544 2.475 2.403 2.328 2.280 2.247 2.204 16 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.538 2.494 2.425 2.352 2.276 2.227 2.194 2.151 17 4.451 3.592 3.197 2.965 2.810 2.699 2.614 2.548 2.494 2.450 2.381 2.308 2.230 2.181 2.148 2.104 18 4.414 3.555 3.160 2.928 2.773 2.661 2.577 2.510 2.456 2.412 2.342 2.269 2.191 2.141 2.107 2.063 19 4.381 3.522 3.127 2.895 2.740 2.628 2.544 2.477 2.423 2.378 2.308 2.234 2.155 2.106 2.071 2.026 20 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348 2.278 2.203 2.124 2.074 2.039 1.994 21 4.325 3.467 3.072 2.840 2.685 2.573 2.488 2.420 2.366 2.321 2.250 2.176 2.096 2.045 2.010 1.965 22 4.301 3.443 3.049 2.817 2.661 2.549 2.464 2.397 2.342 2.297 2.226 2.151 2.071 2.020 1.984 1.938 23 4.279 3.422 3.028 2.796 2.640 2.528 2.442 2.375 2.320 2.275 2.204 2.128 2.048 1.996 1.961 1.914 24 4.260 3.403 3.009 2.776 2.621 2.508 2.423 2.355 2.300 2.255 2.183 2.108 2.027 1.975 1.939 1.892 25 4.242 3.385 2.991 2.759 2.603 2.490 2.405 2.337 2.282 2.236 2.165 2.089 2.007 1.955 1.919 1.872 26 4.225 3.369 2.975 2.743 2.587 2.474 2.388 2.321 2.265 2.220 2.148 2.072 1.990 1.938 1.901 1.853 27 4.210 3.354 2.960 2.728 2.572 2.459 2.373 2.305 2.250 2.204 2.132 2.056 1.974 1.921 1.884 1.836 28 4.196 3.340 2.947 2.714 2.558 2.445 2.359 2.291 2.236 2.190 2.118 2.041 1.959 1.906 1.869 1.820 29 4.183 3.328 2.934 2.701 2.545 2.432 2.346 2.278 2.223 2.177 2.104 2.027 1.945 1.891 1.854 1.806 30 4.171 3.316 2.922 2.690 2.534 2.421 2.334 2.266 2.211 2.165 2.092 2.015 1.932 1.878 1.841 1.792 31 4.160 3.305 2.911 2.679 2.523 2.409 2.323 2.255 2.199 2.153 2.080 2.003 1.920 1.866 1.828 1.779 32 4.149 3.295 2.901 2.668 2.512 2.399 2.313 2.244 2.189 2.142 2.070 1.992 1.908 1.854 1.817 1.767 33 4.139 3.285 2.892 2.659 2.503 2.389 2.303 2.235 2.179 2.133 2.060 1.982 1.898 1.844 1.806 1.756 34 4.130 3.276 2.883 2.650 2.494 2.380 2.294 2.225 2.170 2.123 2.050 1.972 1.888 1.833 1.795 1.745 35 4.121 3.267 2.874 2.641 2.485 2.372 2.285 2.217 2.161 2.114 2.041 1.963 1.878 1.824 1.786 1.735 36 4.113 3.259 2.866 2.634 2.477 2.364 2.277 2.209 2.153 2.106 2.033 1.954 1.870 1.815 1.776 1.726 37 4.105 3.252 2.859 2.626 2.470 2.356 2.270 2.201 2.145 2.098 2.025 1.946 1.861 1.806 1.768 1.717 38 4.098 3.245 2.852 2.619 2.463 2.349 2.262 2.194 2.138 2.091 2.017 1.939 1.853 1.798 1.760 1.708 39 4.091 3.238 2.845 2.612 2.456 2.342 2.255 2.187 2.131 2.084 2.010 1.931 1.846 1.791 1.752 1.700 40 4.085 3.232 2.839 2.606 2.449 2.336 2.249 2.180 2.124 2.077 2.003 1.924 1.839 1.783 1.744 1.693

V-2/v 1 2 3 4 5 6 7 8 9 10 12 15 20 25 30 40 1 4052.181 4999.500 5403.352 5624.583 5763.650 5858.986 5928.356 5981.070 6022.473 6055.847 6106.321 6157.285 6208.730 6239.825 6260.649 6286.782 2 98.503 99.000 99.166 99.249 99.299 99.333 99.356 99.374 99.388 99.399 99.416 99.433 99.449 99.459 99.466 99.474 3 34.116 30.817 29.457 28.710 28.237 27.911 27.672 27.489 27.345 27.229 27.052 26.872 26.690 26.579 26.505 26.411 4 21.198 18.000 16.694 15.977 15.522 15.207 14.976 14.799 14.659 14.546 14.374 14.198 14.020 13.911 13.838 13.745 5 16.258 13.274 12.060 11.392 10.967 10.672 10.456 10.289 10.158 10.051 9.888 9.722 9.553 9.449 9.379 9.291 6 13.745 10.925 9.780 9.148 8.746 8.466 8.260 8.102 7.976 7.874 7.718 7.559 7.396 7.296 7.229 7.143 7 12.246 9.547 8.451 7.847 7.460 7.191 6.993 6.840 6.719 6.620 6.469 6.314 6.155 6.058 5.992 5.908 8 11.259 8.649 7.591 7.006 6.632 6.371 6.178 6.029 5.911 5.814 5.667 5.515 5.359 5.263 5.198 5.116 9 10.561 8.022 6.992 6.422 6.057 5.802 5.613 5.467 5.351 5.257 5.111 4.962 4.808 4.713 4.649 4.567 10 10.044 7.559 6.552 5.994 5.636 5.386 5.200 5.057 4.942 4.849 4.706 4.558 4.405 4.311 4.247 4.165 11 9.646 7.206 6.217 5.668 5.316 5.069 4.886 4.744 4.632 4.539 4.397 4.251 4.099 4.005 3.941 3.860 12 9.330 6.927 5.953 5.412 5.064 4.821 4.640 4.499 4.388 4.296 4.155 4.010 3.858 3.765 3.701 3.619 13 9.074 6.701 5.739 5.205 4.862 4.620 4.441 4.302 4.191 4.100 3.960 3.815 3.665 3.571 3.507 3.425 14 8.862 6.515 5.564 5.035 4.695 4.456 4.278 4.140 4.030 3.939 3.800 3.656 3.505 3.412 3.348 3.266 15 8.683 6.359 5.417 4.893 4.556 4.318 4.142 4.004 3.895 3.805 3.666 3.522 3.372 3.278 3.214 3.132 16 8.531 6.226 5.292 4.773 4.437 4.202 4.026 3.890 3.780 3.691 3.553 3.409 3.259 3.165 3.101 3.018 17 8.400 6.112 5.185 4.669 4.336 4.102 3.927 3.791 3.682 3.593 3.455 3.312 3.162 3.068 3.003 2.920 18 8.285 6.013 5.092 4.579 4.248 4.015 3.841 3.705 3.597 3.508 3.371 3.227 3.077 2.983 2.919 2.835 19 8.185 5.926 5.010 4.500 4.171 3.939 3.765 3.631 3.523 3.434 3.297 3.153 3.003 2.909 2.844 2.761 20 8.096 5.849 4.938 4.431 4.103 3.871 3.699 3.564 3.457 3.368 3.231 3.088 2.938 2.843 2.778 2.695 21 8.017 5.780 4.874 4.369 4.042 3.812 3.640 3.506 3.398 3.310 3.173 3.030 2.880 2.785 2.720 2.636 22 7.945 5.719 4.817 4.313 3.988 3.758 3.587 3.453 3.346 3.258 3.121 2.978 2.827 2.733 2.667 2.583 23 7.881 5.664 4.765 4.264 3.939 3.710 3.539 3.406 3.299 3.211 3.074 2.931 2.781 2.686 2.620 2.535 24 7.823 5.614 4.718 4.218 3.895 3.667 3.496 3.363 3.256 3.168 3.032 2.889 2.738 2.643 2.577 2.492 25 7.770 5.568 4.675 4.177 3.855 3.627 3.457 3.324 3.217 3.129 2.993 2.850 2.699 2.604 2.538 2.453 26 7.721 5.526 4.637 4.140 3.818 3.591 3.421 3.288 3.182 3.094 2.958 2.815 2.664 2.569 2.503 2.417 27 7.677 5.488 4.601 4.106 3.785 3.558 3.388 3.256 3.149 3.062 2.926 2.783 2.632 2.536 2.470 2.384 28 7.636 5.453 4.568 4.074 3.754 3.528 3.358 3.226 3.120 3.032 2.896 2.753 2.602 2.506 2.440 2.354 29 7.598 5.420 4.538 4.045 3.725 3.499 3.330 3.198 3.092 3.005 2.868 2.726 2.574 2.478 2.412 2.325 30 7.562 5.390 4.510 4.018 3.699 3.473 3.304 3.173 3.067 2.979 2.843 2.700 2.549 2.453 2.386 2.299 31 7.530 5.362 4.484 3.993 3.675 3.449 3.281 3.149 3.043 2.955 2.820 2.677 2.525 2.429 2.362 2.275 32 7.499 5.336 4.459 3.969 3.652 3.427 3.258 3.127 3.021 2.934 2.798 2.655 2.503 2.406 2.340 2.252 33 7.471 5.312 4.437 3.948 3.630 3.406 3.238 3.106 3.000 2.913 2.777 2.634 2.482 2.386 2.319 2.231 34 7.444 5.289 4.416 3.927 3.611 3.386 3.218 3.087 2.981 2.894 2.758 2.615 2.463 2.366 2.299 2.211 35 7.419 5.268 4.396 3.908 3.592 3.368 3.200 3.069 2.963 2.876 2.740 2.597 2.445 2.348 2.281 2.193 36 7.396 5.248 4.377 3.890 3.574 3.351 3.183 3.052 2.946 2.859 2.723 2.580 2.428 2.331 2.263 2.175 37 7.373 5.229 4.360 3.873 3.558 3.334 3.167 3.036 2.930 2.843 2.707 2.564 2.412 2.315 2.247 2.159 38 7.353 5.211 4.343 3.858 3.542 3.319 3.152 3.021 2.915 2.828 2.692 2.549 2.397 2.299 2.232 2.143 39 7.333 5.194 4.327 3.843 3.528 3.305 3.137 3.006 2.901 2.814 2.678 2.535 2.382 2.285 2.217 2.128 40 7.314 5.179 4.313 3.828 3.514 3.291 3.124 2.993 2.888 2.801 2.665 2.522 2.369 2.271 2.203 2.114

V .995 .990 .975 .950 .90 .50 .10 .05 .025 .01 .005 1 0.00004 0.00016 0.00098 0.00393 0.01579 0.45494 2.70554 3.84146 5.02389 6.63490 7.87944 2 0.01003 0.02010 0.05064 0.10259 0.21072 1.38629 4.60517 5.99146 7.37776 9.21034 10.5966 3 0.07172 0.11483 0.21580 0.35185 0.58437 2.36597 6.25139 7.81473 9.34840 11.3449 12.8382 4 0.20699 0.29711 0.48442 0.71072 1.06362 3.35669 7.77944 9.48773 11.1433 13.2767 14.8603 5 0.41174 0.55430 0.83121 1.14548 1.61031 4.35146 9.23636 11.0705 12.8325 15.0863 16.7496 6 0.67573 0.87209 1.23734 1.63538 2.20413 5.34812 10.6446 12.5916 14.4494 16.8119 18.5476 7 0.98926 1.23904 1.68987 2.16735 2.83311 6.34581 12.0170 14.0671 16.0128 18.4753 20.2777 8 1.34441 1.64650 2.17973 2.73264 3.48954 7.34412 13.3616 15.5073 17.5345 20.0902 21.9550 9 1.73493 2.08790 2.70039 3.32511 4.16816 8.34283 14.6837 16.9190 19.0228 21.6660 23.5894 10 2.15586 2.55821 3.24697 3.94030 4.86518 9.34182 15.9872 18.3070 20.4832 23.2093 25.1882 11 2.60322 3.05348 3.81575 4.57481 5.57778 10.3410 17.2750 19.6751 21.9200 24.7250 26.7568 12 3.07382 3.57057 4.40379 5.22603 6.30380 11.3403 18.5493 21.0261 23.3367 26.2170 28.2995 13 3.56503 4.10692 5.00875 5.89186 7.04150 12.3398 19.8119 22.3620 24.7356 27.6882 29.8195 14 4.07467 4.66043 5.62873 6.57063 7.78953 13.3393 21.0641 23.6848 26.1189 29.1412 31.3193 15 4.60092 5.22935 6.26214 7.26094 8.54676 14.3389 22.3071 24.9958 27.4884 30.5779 32.8013 16 5.14221 5.81221 6.90766 7.96165 9.31224 15.3385 23.5418 26.2962 28.8454 31.9999 34.2672 17 5.69722 6.40776 7.56419 8.67176 10.0852 16.3382 24.7690 27.5871 30.1910 33.4087 35.7185 18 6.26480 7.01491 8.23075 9.39046 10.8649 17.3379 25.9894 28.8693 31.5264 34.8053 37.1565 19 6.84397 7.63273 8.90652 10.1170 11.6509 18.3377 27.2036 30.1435 32.8523 36.1909 38.5823 20 7.43384 8.26040 9.59078 10.8508 12.4426 19.3374 28.4120 31.4104 34.1696 37.5662 39.9968 21 8.03365 8.89720 10.2829 11.5913 13.2396 20.3372 29.6151 32.6706 35.4789 38.9322 41.4011 22 8.64272 9.54249 10.9823 12.3380 14.0415 21.3370 30.8133 33.9244 36.7807 40.2894 42.7957 23 9.26042 10.1957 11.6886 13.0905 14.8480 22.3369 32.0069 35.1725 38.0756 41.6384 44.1813 24 9.88623 10.8564 12.4012 13.8484 15.6587 23.3367 33.1962 36.4150 39.3641 42.9798 45.5585 25 10.5197 11.5240 13.1197 14.6114 16.4734 24.3366 34.3816 37.6525 40.6465 44.3141 46.9279 26 11.1602 12.1981 13.8439 15.3792 17.2919 25.3365 35.5632 38.8851 41.9232 45.6417 48.2899 27 11.8076 12.8785 14.5734 16.1514 18.1139 26.3363 36.7412 40.1133 43.1945 46.9629 49.6449 28 12.4613 13.5647 15.3079 16.9279 18.9392 27.3362 37.9159 41.3371 44.4608 48.2782 50.9934 29 13.1211 14.2565 16.0471 17.7084 19.7677 28.3361 39.0875 42.5570 45.7223 49.5879 52.3356 30 13.7867 14.9535 16.7908 18.4927 20.5992 29.3360 40.2560 43.7730 46.9792 50.8922 53.6720 31 14.4578 15.6555 17.5387 19.2806 21.4336 30.3359 41.4217 44.9853 48.2319 52.1914 55.0027 32 15.1340 16.3622 18.2908 20.0719 22.2706 31.3359 42.5847 46.1943 49.4804 53.4858 56.3281 33 15.8153 17.0735 19.0467 20.8665 23.1102 32.3358 43.7452 47.3999 50.7251 54.7755 57.6484 34 16.5013 17.7891 19.8063 21.6643 23.9523 33.3357 44.9032 48.6024 51.9660 56.0609 58.9639 35 17.1918 18.5089 20.5694 22.4650 24.7967 34.3356 46.0588 49.8018 53.2033 57.3421 60.2748 36 17.8867 19.2327 21.3359 23.2686 25.6433 35.3356 47.2122 50.9985 54.4373 58.6192 61.5812 37 18.5858 19.9602 22.1056 24.0749 26.4921 36.3355 48.3634 52.1923 55.6680 59.8925 62.8833 38 19.2889 20.6914 22.8785 24.8839 27.3430 37.3355 49.5126 53.3835 56.8955 61.1621 64.1814 39 19.9959 21.4262 23.6543 25.6954 28.1958 38.3354 50.6598 54.5722 58.1201 62.4281 65.4756 40 20.7065 22.1643 24.4330 26.5093 29.0505 39.3353 51.8051 55.7585 59.3417 63.6907 66.7660