Topic: Financial Econometrics Case study

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Please answer:

  1. Tow of three questions.
  2. The answers that you provide should not exceed 500 words for each sub question.
  3. You should use your own words and references in each of your answers provided. Copying expressions from Books or any other source will be considered as unfair practice.
  4. Your answers should be provided in a Word document.

Question B1

Consider the following three models:

(i)         yt  = 0.3yt–1 +  + 0.1yt–2+ut

(ii)        yt  = ut + 0.6ut–1 +0.2ut–2

(iii)       yt = 0.6yt–1 + ut +0.3ut–1

Each of these models can be described as ARMA(p,q). What are p and q in each case? (6 marks)

  1. Discuss shortly the principles behind maximum likelihood estimation. .OLS and maximum likelihood are used to estimate the parameters of a standard linear regression model. Will they give the same estimates? Explain your answer. (10 marks)
    1. Consider the following MA(2)process

𝑋𝑡 = 𝑢𝑡 + 𝜃1𝑢𝑡−1 + 𝜃2𝑢𝑡−2

where 𝑢𝑡is a zero mean white noise process with variance 𝜎2.

  1. Calculate the mean and variance of 𝑋𝑡. (5marks)
  2. Derive the autocorrelation function for this process (5marks)

3)   If 𝜃1= −0.5 and 𝜃2= 0.25, sketch the acf of 𝑋𝑡.        (4marks)

Question B2

  1. Describe briefly the three hypothesis testing procedures that are available under maximum likelihood estimation. Which is likely to be calculated in practice andwhy? (5marks)
  2. What stylized fact of financial data cannot be explained using linear trend models? ( 5marks)
  3. Which of these features can be modelled using a GARCH (1,1) process? ( 5marks)
  4. Why, in empirical research, have researchers preferred GARCH (1,1) models to pure ARCH (p) models? ( 5marks)
  5. Describe two extensions to the original GARCH model. What additional characterisitcs of financial data might they able to capture? ( 5marks)
  6. Consider the following GARCH (1,1)model

𝑦𝑡 = 𝜇 + 𝑢𝑡, 𝑢𝑡~𝑁(0, 𝜎2)

𝜎2= 𝛼0+𝛼𝑢2         + 𝛽𝜎2

𝑡−1                     𝑡−1

if 𝑦𝑡is a financial return series, what range of value are likely the coefficients

𝜇, 𝛼0, 𝛼1 𝑎𝑛𝑑 𝛽 ? (5 marks)

Question B3

  1. Briefly outline Johansen’s methodology for testing for cointegration between a set of variables in the context of VAR. (10 marks)
  2. Explain how the Engle-Granger test for cointegration works. Can this test allow for more than one co-integrating relationship? (10marks)
  3. A researcher uses Johansen’s procedure and obtains the following test statistics (and criticalvalues);
𝑟𝜆𝑚𝑎𝑥95% critical values
038.96233.178
129.14827.169
216.30420.278
38.86114.036
41.9943.962

Determine the number of cointegrating vectors. (10 marks)

Formula sheet

F-tests:

F=                                                                           F =

                               æåe2                              ö

å(yˆ

-y)2                                                       ç       i                                           ÷

Goodness of fit: R2 =

i                      ;            R2 =1-çn-k                    2÷

å(y

– y)2

ç          å(y – y)÷

i                                                                       ç                 i                     ÷

è                n-1     ø

i  

AIC, MAIC:                          ln(åe2/n)+ 2k/n;                              ln(|Sˆ|) + 2k/(T–p)

LR tests:                               t=–2[ln(L1)–ln(L2)];                          t=(T–p)(ln|Sˆ1|–ln|Sˆ2|)

Bera-Jarque:                      X=

nrˆ2+n(rˆ

– 3)2

6 1       24     2

Table 1

Area in the upper tail of the standard normal density function (= a)

0                   z

Example

Z =X -m

s

prob(Z>1.00) = 0.1587 prob(Z>1.96) = 0.0250

z¯®0.000.010.020.030.040.050.060.070.080.09
0.00.50000.49600.49200.48800.48400.48010.47610.47210.46810.4641
0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247
0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859
0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483
0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121
  0.5  0.3085  0.3050  0.3015  0.2981  0.2946  0.2912  0.2877  0.2843  0.2810  0.2776
0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451
0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148
0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867
0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611
  1.0  0.1587  0.1562  0.1539  0.1515  0.1492  0.1469  0.1446  0.1423  0.1401  0.1379
1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170
1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985
1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823
1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681
  1.5  0.0668  0.0655  0.0643  0.0630  0.0618  0.0606  0.0594  0.0582  0.0571  0.0559
1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455
1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367
1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294
1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233
  2.0  0.0228  0.0222  0.0217  0.0212  0.0207  0.0202  0.0197  0.0192  0.0188  0.0183
2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143
2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110
2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084
2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064
  2.5  0.0062  0.0060  0.0059  0.0057  0.0055  0.0054  0.0052  0.0051  0.0049  0.0048
2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036
2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026
2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019
2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014
3.00.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010

Table 2

Critical values for the t distribution for various probability levels (a) and degrees of freedom (n)

0                   t

Example              For n=10, prob(t>1.8125) = 0.1 prob(t>2.6338) =0.025

a® n¯0.250.20.150.10.050.0250.010.0050.0025
11.00001.37641.96263.07776.313712.706231.821063.6559127.321
20.81651.06071.38621.88562.92004.30276.96459.925014.0892
30.76490.97851.24981.63772.35343.18244.54075.84087.4532
40.74070.94101.18961.53322.13182.77653.74694.60415.5975
50.72670.91951.15581.47592.01502.57063.36494.03214.7733
  6  0.7176  0.9057  1.1342  1.4398  1.9432  2.4469  3.1427  3.7074  4.3168
70.71110.89601.11921.41491.89462.36462.99793.49954.0294
80.70640.88891.10811.39681.85952.30602.89653.35543.8325
90.70270.88341.09971.38301.83312.26222.82143.24983.6896
100.69980.87911.09311.37221.81252.22812.76383.16933.5814
  11  0.6974  0.8755  1.0877  1.3634  1.7959  2.2010  2.7181  3.1058  3.4966
120.69550.87261.08321.35621.78232.17882.68103.05453.4284
130.69380.87021.07951.35021.77092.16042.65033.01233.3725
140.69240.86811.07631.34501.76132.14482.62452.97683.3257
150.69120.86621.07351.34061.75312.13152.60252.94673.2860
  16  0.6901  0.8647  1.0711  1.3368  1.7459  2.1199  2.5835  2.9208  3.2520
170.68920.86331.06901.33341.73962.10982.56692.89823.2224
180.68840.86201.06721.33041.73412.10092.55242.87843.1966
190.68760.86101.06551.32771.72912.09302.53952.86093.1737
200.68700.86001.06401.32531.72472.08602.52802.84533.1534
  21  0.6864  0.8591  1.0627  1.3232  1.7207  2.0796  2.5176  2.8314  3.1352
220.68580.85831.06141.32121.71712.07392.50832.81883.1188
230.68530.85751.06031.31951.71392.06872.49992.80733.1040
240.68480.85691.05931.31781.71092.06392.49222.79703.0905
250.68440.85621.05841.31631.70812.05952.48512.78743.0782
  26  0.6840  0.8557  1.0575  1.3150  1.7056  2.0555  2.4786  2.7787  3.0669
270.68370.85511.05671.31371.70332.05182.47272.77073.0565
280.68340.85461.05601.31251.70112.04842.46712.76333.0470
290.68300.85421.05531.31141.69912.04522.46202.75643.0380
300.68280.85381.05471.31041.69732.04232.45732.75003.0298
  40  0.6807  0.8507  1.0500  1.3031  1.6839  2.0211  2.4233  2.7045  2.9712
600.67860.84771.04551.29581.67062.00032.39012.66032.9146
1200.67650.84461.04091.28861.65761.97992.35782.61742.8599
¥0.67450.84161.03641.28161.64491.96002.32642.57592.8071

Table 3

Critical values for the c2 distribution for various

probability levels (a) and degrees offreedom(n)                             0                       X

Example              For n=10, prob(c2>15.987) = 0.1 prob(c2>20.483) =0.025

a® n¯0.9950.990.9750.950.90.750.50.250.10.050.0250.010.005
10.0000.0000.0010.0040.0160.1020.4551.3232.7063.8415.0246.6357.879
20.0100.0200.0510.1030.2110.5751.3862.7734.6055.9917.3789.21010.597
30.0720.1150.2160.3520.5841.2132.3664.1086.2517.8159.34811.34512.838
40.2070.2970.4840.7111.0641.9233.3575.3857.7799.48811.14313.27714.860
50.4120.5540.8311.1451.6102.6754.3516.6269.23611.07012.83215.08616.750
  6  0.676  0.872  1.237  1.635  2.204  3.455  5.348  7.841  10.645  12.592  14.449  16.812  18.548
70.9891.2391.6902.1672.8334.2556.3469.03712.01714.06716.01318.47520.278
81.3441.6472.1802.7333.4905.0717.34410.21913.36215.50717.53520.09021.955
91.7352.0882.7003.3254.1685.8998.34311.38914.68416.91919.02321.66623.589
102.1562.5583.2473.9404.8656.7379.34212.54915.98718.30720.48323.20925.188
  11  2.603  3.053  3.816  4.575  5.578  7.584  10.341  13.701  17.275  19.675  21.920  24.725  26.757
123.0743.5714.4045.2266.3048.43811.34014.84518.54921.02623.33726.21728.300
133.5654.1075.0095.8927.0419.29912.34015.98419.81222.36224.73627.68829.819
144.0754.6605.6296.5717.79010.16513.33917.11721.06423.68526.11929.14131.319
154.6015.2296.2627.2618.54711.03714.33918.24522.30724.99627.48830.57832.801
  16  5.142  5.812  6.908  7.962  9.312  11.912  15.338  19.369  23.542  26.296  28.845  32.000  34.267
175.6976.4087.5648.67210.08512.79216.33820.48924.76927.58730.19133.40935.718
186.2657.0158.2319.39010.86513.67517.33821.60525.98928.86931.52634.80537.156
196.8447.6338.90710.11711.65114.56218.33822.71827.20430.14432.85236.19138.582
207.4348.2609.59110.85112.44315.45219.33723.82828.41231.41034.17037.56639.997
  21  8.034  8.897  10.283  11.591  13.240  16.344  20.337  24.935  29.615  32.671  35.479  38.932  41.401
228.6439.54210.98212.33814.04117.24021.33726.03930.81333.92436.78140.28942.796
239.26010.19611.68913.09114.84818.13722.33727.14132.00735.17238.07641.63844.181
249.88610.85612.40113.84815.65919.03723.33728.24133.19636.41539.36442.98045.558
2510.52011.52413.12014.61116.47319.93924.33729.33934.38237.65240.64644.31446.928
  30  13.787  14.953  16.791  18.493  20.599  24.478  29.336  34.800  40.256  43.773  46.979  50.892  53.672
4020.70722.16424.43326.50929.05133.66039.33545.61651.80555.75859.34263.69166.766
5027.99129.70732.35734.76437.68942.94249.33556.33463.16767.50571.42076.15479.490
6035.53437.48540.48243.18846.45952.29459.33566.98174.39779.08283.29888.37991.952
7043.27545.44248.75851.73955.32961.69869.33477.57785.52790.53195.021100.43104.22
  80  51.172  53.540  57.153  60.391  64.278  71.145  79.334  88.130  96.578  101.88  106.63  112.33  116.32
9059.19661.75465.64769.12673.29180.62589.33498.650107.57113.15118.14124.12128.30
10067.32870.06574.22277.92982.35890.13399.334109.14118.50124.34129.56135.81140.17
Z-2.576-2.326-1.960-1.645-1.282-0.6740.0000.6741.2821.6451.9602.3262.576

Note: For n>100, use c2

=1 (z +

2

2n-1)2

, where z = standardized normal deviate shown in the

bottom row of the table.

Table4a                Critical values for the F distribution, a=0.01, degrees of freedom = n1,n2

Example         For n1=4, n2=20, prob(F>4.43) =0.01

n1® n2¯123456789101215203050100200500¥
14052499954045624576458595928598160226056610761576209626063026334635063606366
298.599.099.299.399.399.399.499.499.499.499.499.499.499.599.599.599.599.599.5
334.130.829.528.728.227.927.727.527.327.227.126.926.726.526.426.226.226.126.1
421.218.016.716.015.515.215.014.814.714.514.414.214.013.813.713.613.513.513.5
516.313.312.111.411.010.710.510.310.210.19.899.729.559.389.249.139.089.049.02
613.710.99.789.158.758.478.268.107.987.877.727.567.407.237.096.996.936.906.88
712.29.558.457.857.467.196.996.846.726.626.476.316.165.995.865.755.705.675.65
811.38.657.597.016.636.376.186.035.915.815.675.525.365.205.074.964.914.884.86
910.68.026.996.426.065.805.615.475.355.265.114.964.814.654.524.414.364.334.31
1010.07.566.555.995.645.395.205.064.944.854.714.564.414.254.124.013.963.933.91
119.657.216.225.675.325.074.894.744.634.544.404.254.103.943.813.713.663.623.60
129.336.935.955.415.064.824.644.504.394.304.164.013.863.703.573.473.413.383.36
139.076.705.745.214.864.624.444.304.194.103.963.823.663.513.383.273.223.193.17
148.866.515.565.044.694.464.284.144.033.943.803.663.513.353.223.113.063.033.00
158.686.365.424.894.564.324.144.003.893.803.673.523.373.213.082.982.922.892.87
168.536.235.294.774.444.204.033.893.783.693.553.413.263.102.972.862.812.782.75
178.406.115.194.674.344.103.933.793.683.593.463.313.163.002.872.762.712.682.65
188.296.015.094.584.254.013.843.713.603.513.373.233.082.922.782.682.622.592.57
198.185.935.014.504.173.943.773.633.523.433.303.153.002.842.712.602.552.512.49
208.105.854.944.434.103.873.703.563.463.373.233.092.942.782.642.542.482.442.42
218.025.784.874.374.043.813.643.513.403.313.173.032.882.722.582.482.422.382.36
227.955.724.824.313.993.763.593.453.353.263.122.982.832.672.532.422.362.332.31
237.885.664.764.263.943.713.543.413.303.213.072.932.782.622.482.372.322.282.26
247.825.614.724.223.903.673.503.363.263.173.032.892.742.582.442.332.272.242.21
257.775.574.684.183.853.633.463.323.223.132.992.852.702.542.402.292.232.192.17
267.725.534.644.143.823.593.423.293.183.092.962.812.662.502.362.252.192.162.13
277.685.494.604.113.783.563.393.263.153.062.932.782.632.472.332.222.162.122.10
287.645.454.574.073.753.533.363.233.123.032.902.752.602.442.302.192.132.092.06
297.605.424.544.043.733.503.333.203.093.002.872.732.572.412.272.162.102.062.03
307.565.394.514.023.703.473.303.173.072.982.842.702.552.392.252.132.072.032.01
357.425.274.403.913.593.373.203.072.962.882.742.602.442.282.142.021.961.921.96
407.315.184.313.833.513.293.122.992.892.802.662.522.372.202.061.941.871.831.80
457.235.114.253.773.453.233.072.942.832.742.612.462.312.142.001.881.811.771.78
507.175.064.203.723.413.193.022.892.782.702.562.422.272.101.951.821.761.711.68
607.084.984.133.653.343.122.952.822.722.632.502.352.202.031.881.751.681.631.60
707.014.924.073.603.293.072.912.782.672.592.452.312.151.981.831.701.621.571.54
806.964.884.043.563.263.042.872.742.642.552.422.272.121.941.791.651.581.531.49
1006.904.823.983.513.212.992.822.692.592.502.372.222.071.891.741.601.521.471.43
1506.814.753.913.453.142.922.762.632.532.442.312.162.001.831.661.521.431.381.33
2006.764.713.883.413.112.892.732.602.502.412.272.131.971.791.631.481.391.331.28
4006.704.663.833.373.062.852.682.562.452.372.232.081.921.751.581.421.321.251.19
10006.664.633.803.343.042.822.662.532.432.342.202.061.901.721.541.381.281.191.11
¥6.634.613.783.323.022.802.642.512.412.322.182.041.881.701.521.361.251.151.00

Table4b          Critical values for the F distribution, a=0.025, degrees of freedom = n1,n2

Example         For n1=4, n2=20, prob(F>3.51) =0.025

n1® n2¯123456789101215203050100200500¥
1648799864900922937948957963969977985993100110081013101610171018
238.539.039.239.239.339.339.439.439.439.439.439.439.439.539.539.539.539.539.5
317.416.015.415.114.914.714.614.514.514.414.314.314.214.114.014.013.913.913.9
412.210.610.09.609.369.209.078.988.908.848.758.668.568.468.388.328.298.278.26
510.08.437.767.397.156.986.856.766.686.626.526.436.336.236.146.086.056.036.02
68.817.266.606.235.995.825.705.605.525.465.375.275.175.074.984.924.884.864.85
78.076.545.895.525.295.124.994.904.824.764.674.574.474.364.284.214.184.164.14
87.576.065.425.054.824.654.534.434.364.304.204.104.003.893.813.743.703.683.67
97.215.715.084.724.484.324.204.104.033.963.873.773.673.563.473.403.373.353.33
106.945.464.834.474.244.073.953.853.783.723.623.523.423.313.223.153.123.093.08
116.725.264.634.284.043.883.763.663.593.533.433.333.233.123.032.962.922.902.88
126.555.104.474.123.893.733.613.513.443.373.283.183.072.962.872.802.762.742.72
136.414.974.354.003.773.603.483.393.313.253.153.052.952.842.742.672.632.612.60
146.304.864.243.893.663.503.383.293.213.153.052.952.842.732.642.562.532.502.49
156.204.774.153.803.583.413.293.203.123.062.962.862.762.642.552.472.442.412.40
166.124.694.083.733.503.343.223.123.052.992.892.792.682.572.472.402.362.332.32
176.044.624.013.663.443.283.163.062.982.922.822.722.622.502.412.332.292.262.25
185.984.563.953.613.383.223.103.012.932.872.772.672.562.442.352.272.232.202.19
195.924.513.903.563.333.173.052.962.882.822.722.622.512.392.302.222.182.152.13
205.874.463.863.513.293.133.012.912.842.772.682.572.462.352.252.172.132.102.09
215.834.423.823.483.253.092.972.872.802.732.642.532.422.312.212.132.092.062.04
225.794.383.783.443.223.052.932.842.762.702.602.502.392.272.172.092.052.022.00
235.754.353.753.413.183.022.902.812.732.672.572.472.362.242.142.062.011.991.97
245.724.323.723.383.152.992.872.782.702.642.542.442.332.212.112.021.981.951.94
255.694.293.693.353.132.972.852.752.682.612.512.412.302.182.082.001.951.921.91
265.664.273.673.333.102.942.822.732.652.592.492.392.282.162.051.971.921.901.88
275.634.243.653.313.082.922.802.712.632.572.472.362.252.132.031.941.901.871.85
285.614.223.633.293.062.902.782.692.612.552.452.342.232.112.011.921.881.851.83
295.594.203.613.273.042.882.762.672.592.532.432.322.212.091.991.901.861.831.81
305.574.183.593.253.032.872.752.652.572.512.412.312.202.071.971.881.841.811.79
355.484.113.523.182.962.802.682.582.502.442.342.232.122.001.891.801.751.721.70
405.424.053.463.132.902.742.622.532.452.392.292.182.071.941.831.741.691.661.64
455.384.013.423.092.862.702.582.492.412.352.252.142.031.901.791.691.641.611.59
505.343.973.393.052.832.672.552.462.382.322.222.111.991.871.751.661.601.571.55
605.293.933.343.012.792.632.512.412.332.272.172.061.941.821.701.601.541.511.48
705.253.893.312.972.752.592.472.382.302.242.142.031.911.781.661.561.501.461.44
805.223.863.282.952.732.572.452.352.282.212.112.001.881.751.631.531.471.431.40
1005.183.833.252.922.702.542.422.322.242.182.081.971.851.711.591.481.421.381.35
1505.133.783.202.872.652.492.372.282.202.132.031.921.801.671.541.421.351.311.27
2005.103.763.182.852.632.472.352.262.182.112.011.901.781.641.511.391.321.271.23
4005.063.723.152.822.602.442.322.222.152.081.981.871.741.601.471.351.271.211.15
10005.043.703.132.802.582.422.302.202.132.061.961.851.721.581.451.321.231.161.09
¥5.023.693.122.792.572.412.292.192.112.051.941.831.711.571.431.301.211.131.00

Table 4c          Critical values for the F distribution, a=0.05, degrees of freedom = n1,n2

Example         For n1=4, n2=20, prob(F>2.87) =0.05

n1® n2¯123456789101215203050100200500¥
1161199216225230234237239241242244246248250252253254254254
218.519.019.219.219.319.319.419.419.419.419.419.419.419.519.519.519.519.519.5
310.19.559.289.129.018.948.898.858.818.798.748.708.668.628.588.558.548.538.53
47.716.946.596.396.266.166.096.046.005.965.915.865.805.755.705.665.655.645.63
56.615.795.415.195.054.954.884.824.774.744.684.624.564.504.444.414.394.374.37
65.995.144.764.534.394.284.214.154.104.064.003.943.873.813.753.713.693.683.67
75.594.744.354.123.973.873.793.733.683.643.573.513.443.383.323.273.253.243.23
85.324.464.073.843.693.583.503.443.393.353.283.223.153.083.022.972.952.942.93
95.124.263.863.633.483.373.293.233.183.143.073.012.942.862.802.762.732.722.71
104.964.103.713.483.333.223.143.073.022.982.912.852.772.702.642.592.562.552.54
114.843.983.593.363.203.093.012.952.902.852.792.722.652.572.512.462.432.422.40
124.753.893.493.263.113.002.912.852.802.752.692.622.542.472.402.352.322.312.30
134.673.813.413.183.032.922.832.772.712.672.602.532.462.382.312.262.232.222.21
144.603.743.343.112.962.852.762.702.652.602.532.462.392.312.242.192.162.142.13
154.543.683.293.062.902.792.712.642.592.542.482.402.332.252.182.122.102.082.07
164.493.633.243.012.852.742.662.592.542.492.422.352.282.192.122.072.042.022.01
174.453.593.202.962.812.702.612.552.492.452.382.312.232.152.082.021.991.971.96
184.413.553.162.932.772.662.582.512.462.412.342.272.192.112.041.981.951.931.92
194.383.523.132.902.742.632.542.482.422.382.312.232.162.072.001.941.911.891.88
204.353.493.102.872.712.602.512.452.392.352.282.202.122.041.971.911.881.861.84
214.323.473.072.842.682.572.492.422.372.322.252.182.102.011.941.881.841.831.81
224.303.443.052.822.662.552.462.402.342.302.232.152.071.981.911.851.821.801.78
234.283.423.032.802.642.532.442.372.322.272.202.132.051.961.881.821.791.771.76
244.263.403.012.782.622.512.422.362.302.252.182.112.031.941.861.801.771.751.73
254.243.392.992.762.602.492.402.342.282.242.162.092.011.921.841.781.751.731.71
264.233.372.982.742.592.472.392.322.272.222.152.071.991.901.821.761.731.711.69
274.213.352.962.732.572.462.372.312.252.202.132.061.971.881.811.741.711.691.67
284.203.342.952.712.562.452.362.292.242.192.122.041.961.871.791.731.691.671.65
294.183.332.932.702.552.432.352.282.222.182.102.031.941.851.771.711.671.651.64
304.173.322.922.692.532.422.332.272.212.162.092.011.931.841.761.701.661.641.62
354.123.272.872.642.492.372.292.222.162.112.041.961.881.791.701.631.601.571.56
404.083.232.842.612.452.342.252.182.122.082.001.921.841.741.661.591.551.531.51
454.063.202.812.582.422.312.222.152.102.051.971.891.811.711.631.551.511.491.47
504.033.182.792.562.402.292.202.132.072.031.951.871.781.691.601.521.481.461.44
604.003.152.762.532.372.252.172.102.041.991.921.841.751.651.561.481.441.411.39
703.983.132.742.502.352.232.142.072.021.971.891.811.721.621.531.451.401.371.35
803.963.112.722.492.332.212.132.062.001.951.881.791.701.601.511.431.381.351.32
1003.943.092.702.462.312.192.102.031.971.931.851.771.681.571.481.391.341.311.28
1503.903.062.662.432.272.162.072.001.941.891.821.731.641.541.441.341.291.251.22
2003.893.042.652.422.262.142.061.981.931.881.801.721.621.521.411.321.261.221.19
4003.863.022.632.392.242.122.031.961.901.851.781.691.601.491.381.281.221.171.13
10003.853.002.612.382.222.112.021.951.891.841.761.681.581.471.361.261.191.131.08
¥3.843.002.602.372.212.102.011.941.881.831.751.671.571.461.351.241.171.111.00

Table4d          Critical values for the F distribution, a=0.1, degrees of freedom = n1,n2

Example         For n1=4, n2=20, prob(F>2.25) =0.1

n1® n2¯123456789101215203050100200500¥
139.949.553.655.857.258.258.959.459.960.260.761.261.762.362.763.063.263.363.3
28.539.009.169.249.299.339.359.379.389.399.419.429.449.469.479.489.499.499.49
35.545.465.395.345.315.285.275.255.245.235.225.205.185.175.155.145.145.145.13
44.544.324.194.114.054.013.983.953.943.923.903.873.843.823.803.783.773.763.76
54.063.783.623.523.453.403.373.343.323.303.273.243.213.173.153.133.123.113.11
63.783.463.293.183.113.053.012.982.962.942.902.872.842.802.772.752.732.732.72
73.593.263.072.962.882.832.782.752.722.702.672.632.592.562.522.502.482.482.47
83.463.112.922.812.732.672.622.592.562.542.502.462.422.382.352.322.312.302.29
93.363.012.812.692.612.552.512.472.442.422.382.342.302.252.222.192.172.172.16
103.292.922.732.612.522.462.412.382.352.322.282.242.202.162.122.092.072.062.06
113.232.862.662.542.452.392.342.302.272.252.212.172.122.082.042.011.991.981.97
123.182.812.612.482.392.332.282.242.212.192.152.102.062.011.971.941.921.911.90
133.142.762.562.432.352.282.232.202.162.142.102.052.011.961.921.881.861.851.85
143.102.732.522.392.312.242.192.152.122.102.052.011.961.911.871.831.821.801.80
153.072.702.492.362.272.212.162.122.092.062.021.971.921.871.831.791.771.761.76
163.052.672.462.332.242.182.132.092.062.031.991.941.891.841.791.761.741.731.72
173.032.642.442.312.222.152.102.062.032.001.961.911.861.811.761.731.711.691.69
183.012.622.422.292.202.132.082.042.001.981.931.891.841.781.741.701.681.671.66
192.992.612.402.272.182.112.062.021.981.961.911.861.811.761.711.671.651.641.63
202.972.592.382.252.162.092.042.001.961.941.891.841.791.741.691.651.631.621.61
212.962.572.362.232.142.082.021.981.951.921.871.831.781.721.671.631.611.601.59
222.952.562.352.222.132.062.011.971.931.901.861.811.761.701.651.611.591.581.57
232.942.552.342.212.112.051.991.951.921.891.841.801.741.691.641.591.571.561.55
242.932.542.332.192.102.041.981.941.911.881.831.781.731.671.621.581.561.541.53
252.922.532.322.182.092.021.971.931.891.871.821.771.721.661.611.561.541.531.52
262.912.522.312.172.082.011.961.921.881.861.811.761.711.651.591.551.531.511.50
272.902.512.302.172.072.001.951.911.871.851.801.751.701.641.581.541.521.501.49
282.892.502.292.162.062.001.941.901.871.841.791.741.691.631.571.531.501.491.48
292.892.502.282.152.061.991.931.891.861.831.781.731.681.621.561.521.491.481.47
302.882.492.282.142.051.981.931.881.851.821.771.721.671.611.551.511.481.471.46
352.852.462.252.112.021.951.901.851.821.791.741.691.631.571.511.471.441.421.41
402.842.442.232.092.001.931.871.831.791.761.711.661.611.541.481.431.411.391.38
452.822.422.212.071.981.911.851.811.771.741.701.641.581.521.461.411.381.361.35
502.812.412.202.061.971.901.841.801.761.731.681.631.571.501.441.391.361.341.33
602.792.392.182.041.951.871.821.771.741.711.661.601.541.481.411.361.331.311.29
702.782.382.162.031.931.861.801.761.721.691.641.591.531.461.391.341.301.281.27
802.772.372.152.021.921.851.791.751.711.681.631.571.511.441.381.321.281.261.24
1002.762.362.142.001.911.831.781.731.691.661.611.561.491.421.351.291.261.231.21
1502.742.342.121.981.891.811.761.711.671.641.591.531.471.401.331.261.221.191.17
2002.732.332.111.971.881.801.751.701.661.631.581.521.461.381.311.241.201.171.14
4002.722.322.101.961.861.791.731.691.651.611.561.501.441.361.291.211.171.131.10
10002.712.312.091.951.851.781.721.681.641.611.551.491.431.351.271.201.151.101.06
¥2.712.302.081.941.851.771.721.671.631.601.551.491.421.341.261.191.131.081.00

Table 5 Dickey-Fuller critical values for different significance levels, 𝜶

Sample size T0.010.0250.050.1
  τ (no constant, no trend) 
25−2.66−2.26−1.95−1.60
50−2.62−2.25−1.95−1.61
100−2.60−2.24−1.95−1.61
250−2.58−2.23−1.95−1.62
500−2.58−2.23−1.95−1.62
−2.58−2.23−1.95−1.62
  τμ (constant, no trend) 
25−3.75−3.33−3.00−2.63
50−3.58−3.22−2.93−2.60
100−3.51−3.17−2.89−2.58
250−3.46−3.14−2.88−2.57
500−3.44−3.13−2.87−2.57
−3.43−3.12−2.86−2.57
  ττ (constant, trend) 
25−4.38−3.95−3.60−3.24
50−4.15−3.80−3.50−3.18
100−4.04−3.73−3.45−3.15
250−3.99−3.69−3.43−3.13
500−3.98−3.68−3.42−3.13
−3.96−3.66−3.41−3.12

Table 6 Critical values for the Engle-Granger cointegration test on regression residuals with no constant in test regression for different significance levels, 𝜶

Number of variables in systemSample size T0.010.050.1
    250−4.32−3.67−3.28
100−4.07−3.37−3.03
200−4.00−3.37−3.02
     50    −4.84    −4.11    −3.73
3100−4.45−3.93−3.59
 200−4.35−3.78−3.47
   50  −4.94  −4.35  −4.02
4100−4.75−4.22−3.89
 200−4.70−4.18−3.89
   50  −5.41  −4.76  −4.42
5100−5.18−4.58−4.26
 200−5.02−4.48−4.18

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