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Empirical Finance :Analysis of non-stationary processes II: long-run relationships in empirical finance
Module Leader: Dr Stuart Fraser
stuart.fraser@wbs.ac.uk
Room D1.18 (Social Studies)Warwick Business School 2
Today
Multivariate analysis with nonstationaryvariables
1. Spurious regressions
2. Cointegration
3. Analyzing long-run relationships in empirical finance
Seminar 7: Cointegrationanalysis of long-run PPP.
(Brooks Chps7.4-7.8; VerbeekChp9)Warwick Business School 3
Spurious regressions
The problem: Independentnon-stationary processes can appear to be related:
For example suppose…
…then a regression of Y on X…
…will typically display:
–A high R-sq (even though it should be zero).
–A significant t-stat for β(even though it should be insignificant).
Clearly this can be very misleading in empirical work. However the regression also contains a clue that there is a problem with the equation:
–The error term is highly autocorrelated(in fact it’s I(1) –see below).
()()().0,cov ,,0~ ,,0~ ,212222121111=+=+=−−ttttttttttIIDXXIIDYYεεσεεσεε
Y and X are independent random walks
(I(1) processes).
tttXYεβμ++=Warwick Business School 4
Spurious regression: sampling distributions of the
t-stat and R-sq from a simulation with 100,000 samples of
independent random walks with T=10,0000200040006000800010000120001400016000-500-375-250-1250125250375Series: T_SPURSample 1 100000Observations 100000Mean 0.023541Median 0.088570Maximum 448.0761Minimum -480.9837Std. Dev. 74.15763Skewness -0.033843Kurtosis 3.963160Jarque-Bera 3884.412Probability 0.0000000400080001200016000200000.000.250.500.75Series: RSQ_SPURSample 1 100000Observations 100000Mean 0.241821Median 0.172911Maximum 0.958967Minimum 1.78e-12Std. Dev. 0.226885Skewness 0.843509Kurtosis 2.674488Jarque-Bera 12299.95Probability 0.000000If we run regressions with independent seriesthen we’d expect the absolute value ofthe t-stat for βto be >1.96 only 5%of the timein repeated samples (based on a normal dist.).However in this case the distribution of t is highlynon-normal:in fact 97.6% of the t-values are greater than 1.96 in absolute value! The t-ratios in these regression tend to indicate that there is asignificant relationship even though there is none.We’d expect the R-sq to be close to 0since the series are independent.However here we find that: •R-sq>0.5 in 16.4% of the samples.•R-sq>0.9 in 0.1% of the samples!Warwick Business School 5
Spurious regression: simulation results continued
AutocorrelationPartial CorrelationAC PAC Q-Stat Prob |******** |******10.9980.9989867.40 |******** | 20.9960.01197000 |******** | 30.995-0.003294980 |******** | 40.9930.007392610 |******** | 50.9910.006489910 |******** | 60.989-0.011586860 |******** | 70.9870.001683470 |******** | 80.9860.008779750 |******** | 90.984-0.02875680 |******** | 100.9820.015971270 |******** | 110.980.0131066540 |******** | 120.979-0.0131161470 |******** | 130.97701256070 |*******| | 140.975-0.0081350330 |*******| | 150.9730.0061444260 |*******| | 160.971-0.0081537860 |*******| | 170.970.0051631120 |*******| | 180.9680.011724050 |*******| | 190.9660.0031816670 |*******| | 200.964-0.0021908960 |*******| | 210.9630.0042000940 |*******| | 220.9610.0112092610 |*******| | 230.959-0.0192183960 |*******| | 240.9580.0122275000
Sample ACF of residuals
The residuals are highly autocorrelated.
In fact they look non-stationary!
This is a maj