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Home , Cricket, Over (cricket), Result (cricket), Run (cricket), Run rate, Test cricket

Evolution of Test in Last Six Decades A Univariate Time Series Analysis

Mayank Nagpal Sumit Mishra

1 Introduction

We intend to analyse the structural changes in the average annual -rate, i.e., how many runs are scored in each , a measure of how much bat dominates the or how aggressively teams bat. is the traditional format of the . It is considered to be a snail-form of the game when we compare it with newer versions of the game, viz, ODI and T20 .A test cricket match lasting 5 days is apparently a lot less exciting for some than an ODI which lasts for eight hours or a T20 match which is matter of two-three hours. The general view is that with the advent of new technology, pitches that are more batsmen-friendly and craving for in each game, the average seems to have increased. Some analysts attribute this change to the emergence of newer formats and other innovations in the game. The question we are trying to answer is about whether these factors like those mentioned below had any significant impact the game.

The events whose effect we like to capture are: • Advent of (ODI):With dying popularity of test cricket matches during 1960s,a tournament called Gillette Cup was played in 1963 in . The cup had sixty- five overs a side matches. This tourney was a knockout one and it became quite popular and laid foundation for a sleeker format of the game known as ODI-fifty overs a side game. The advent of ODI in 1971 is considered to be a turning point in the history of the game; • circus: Kerry Packer was an Australian media baron who formed a rebel league of cricket known as in 1977 which changed the nature of the game being played forever; • In 2003, a new and vibrant Twenty 20 format was introduced.The format of closely resembles a form of the game that has been popular in English amateur cricket since 1960.It is expected to increase interest in the game by pulling more people towards cricket as a whole. In recent times with high scores being chased easily in all formats of the game, it is being said that this is due to the effect of T20 on other formats of the game. We plan to model and empirically test for this effect.

We apply time series analysis to answer the question.We perform a similar analysis for two other series i.e. the proportion of matches drawn in a year and the variance of the run rate for a year.

2 Data

The annual data for the paper is downloaded from the ESPN Cricinfo website. We use annual data for average run rate, variance of run rates for the year and the proportion of matches drawn for every year from 1946. Although we had data from the year 1877, we use only the post World War-II data as very few matches were played before this period leaving us with only 63 data points.

1 As there are factors other than what we plan to model affecting the runs per over series such as the weather conditions, type of teams playing, condition on the , etc., an by innings study of the series would not have been feasible. To nullify the effect we converted the data into an annual series by taking the average for an entire year. As matches are played throughout the year and in all weather conditions, we hope to nullify the effects of these factors by averaging out the data. Also, there are other factors such as the changing nature of the pitches and the kind of equipment used which might affect the average runs per over in a year. We plan to perform this time series analysis for test matches as well as for one day internationals and study whether different events have affected the game and how cricket has changed over the period of 60 years.

Table-I

Summary Statistics

Run Rate Variance Proportion Of Matches Drawn Mean 2.78 0.66 38.23 Median 2.77 0.49 36.11 Standard Error 0.04 0.12 1.79 Standard Deviation 0.33 1 14.2 Sample Variance 0.11 1 201.7 Kurtosis -0.47 52.97 -0.33 Skewness 0.1 7.02 0.54 Range 1.44 7.98 63.89 Minimum 2.04 0.19 11.11 Maximum 3.48 8.17 75 Sum 174.86 41.71 2408.77 Data Points 63 63 63

3 Analysis of Run-Rate Series

The time plot suggests that the average annual run rate remained lower than the average till 1976 after which it hovered around the average peaking rarely until 2001. There seems to be a positive trend in the series, i.e. the value seems to increase with time. In post 2001 period the average run rate has remained pretty high(consistently above 3).It seems that there is a structural change in the run-rate in late 1950s and mid-1960s,late 1970s and then in 2000. We will apply proper analytical tests for this hypothesis to ascertain whether structural break exists.

2 Time Plot of Run Rate

ACF&PACF Plot for Run Rate Series

There appears to be a clear dependence on past values looking at the ACF and PACF of the series. Also, looking at the time plot there appears to be a dependence on time for the run rate series. The series appears to be non-stationary as there is a slow decay in the value of the ACFs.To test for a stationary trend in the series,we regress run rate on time and get the following result.

3 Dependent Variable:Run Rate

Independent Variable Estimate Standard Error p-value Intercept* 2.281589 0.042254 2e-16 Time* 0.015438 0.001148 2e-16 The regression confirms our suspicion that there is a trend in the run-rate series.We would be testing, using the structural break test, whether the introduction of the new formats made any significant difference in the rise.

The positive coefficient on time indicates a positive trend in the series. We remove this trend by detrending the data and get the new series which looks like what we have plotted in the figure below. ACF and PACF of this detrended series are also stacked along with the given time plot.

Looking at the plots of the ACF are PACF we can predict the models for the de-trended run rate series to be either AR (1), AR(4) or ARMA(2,1). To find the best fitting model we calculate the AIC, SBC for various order ARMA models and test if the residuals for these models are white noise. This test is essential for our analysis as any model which does not have white noise error terms is said to be a misspecified model. The results show that the model with lowest values of SBC and AIC is AR(1).

Dependent Variable:Detrended Run Rate[AR(1)]

Independent Variable Estimate Standard Error Intercept 0.0031 0.0370 AR (1) [t−1∗] 0.5442 0.1042 To test for GARCH (Generalised Auto Regressive Conditional Heteroscedasticity) effects for the run-rate series we use the residuals and residual squared series of the model and plot these errors along with their ACFs and PACFs.

4 The ACF and PACF plots of the residual squared term do not point to any ARCH/GARCH effects in the series. Moreover, when we ran the test for SBC and AIC values (Given in the Ap- pendix), ARMA (0,0) turned out to be the DGP that represents the residual squared series.

Thus, the estimated model for the run-rate series is as given below:

Xt = 2.28 + 0.015t + φ(L)t + ωt (1) where φ(L) = 1 − 0.54L (2) and ωt ∼ N(0, 0.019) (3)

Xt = 2.28 + 0.015t + 0.54t−1 + t (4)

The equation implies that the run rate for a year depends on the previous years innovations, i.e. if for some reason the run rate for some year is higher than expected or higher than the normal run rate due to some innovations in the game, it would be high for the next year as well. Thus, if any innovation (for instance new and improved bats) introduced in the game causes an increase in the rate of runs scored,there will be persistence. To answer our question that whether structural break exists, we need to test the hypothesis that there is no structural break at the specified points in the series. We perform the Chow test to for structural breaks in the three series. For the first series we calculate the F statistic and plot it for all values. We perform the test separately for the regression of Run Rate on time and for the AR(1) model of the de-trended series. The red lines represent the critical values for a 95% confidence interval. It can be seen that the F statistic is greater than the critical value for time 10 to 16 i.e. from the years 1956 to 1962, reaching a peak at time =12, i.e. 1958. The plot suggests that the Chow test accepts (does not reject) our hypothesis of no structural breaks in the seasons 1970-71, 1977-78 and 2003-04.For the detrended run-rate series, the F-statistics values sit perfectly below the red line indicating absence of any structural break in the data.

5 F statistics for Time-trend and De-trended AR(1) Model For Run-Rate Series

4 Analysis of Volatility in Run-Rate

To test for the effects of these factors, we also study the annual series of the variance of run rates. The variance series captures the variability in the conditions in which cricket is played and also the adaptability of the players to foreign conditions when they tour other countries.The figure below suggests that the volatility in the average run rate has remained quiescent for the whole time-span except for in 1992 when it shot up to 8.2.The common view is that the variance of the run rate taken over the period of a year has decreased recently due to kind of pitches that are being prepared. Unlike the decades of 70s and 80s when the pitches on which the matches being played were heterogeneous, i.e., at some grounds the pitches would be more bowler-friendly and some would be batsman friendly, the pitches prepared over all countries in past fifteen years or so have become more or less similar to each other resulting in a fall in the variation in the run rate. This might also be due to fact that globalization and the fact that more and more players travel to other countries and continents, the ease with which players get adapted to the kind of conditions encountered in different countries very easily. But the change has come very recently and thus such a scant of observations might just not be enough to perform an analysis on this series and capture the effects of the changing times.

6 As the presence of the outlier might affect our analysis, we will perform all our tests and analysis without the outlier, i.e. without the value for the year 1992. Following figure shows the time plot of the volatility in run-rate excluding the outlier.This changes the behavior of ACFs and PACFs significantly.The competing models for the series are AR(1),MA(1)and MA(2).

When we regressed this series on time, the coefficient turned out to be significant at 95% confidence

7 interval. The result is summarized below.

Independent Variable Estimate Standard Error p-value Intercept 0.3577 0.068 0.000 Time 0.00581 0.0018 0.003 We plot the ACFs and PACFs of the detrended volatility series and compare them with those of the volatility series. In addition to the coefficient on time being significant detrending seems to have significantly changed the ACFs and PACFs for the series. We know that if we detrend a series in which the deterministic trend is actually absent ,an MA 1 term is introduced in the series. This is exactly what happens in our detrended series, i.e. we get an MA(1) term on modeling the series using the AIC and SBC criteria. Thus we reject the hypothesis of a deterministic trend in the volatility series even though the regression results and the correlation plots indicate so. Moreover, closer analysis of the time plot of variance suggests it to be a stationary time series with constant mean and variance over time.

Based on the AIC and SBC values of different models,AR(1) is the DGP of the volatility series.The suggested model is:

yt = 0.5377 + 0.349yt−1 + t (5) where t ∼ N(0, 1) (6)

The plots of residual and residual squared series alongwith the ACFs and the PACFs are given be- low.Based on the SBC and AIC values,the DGP that best describes the series is ARMA(0,0).Hence,there is no ARCH/GARCH model involved in this series.

8 To test for structural breaks in the series, we need to test the hypothesis that there is no structural break at the specified points in the series. We perform the Chow test to look for structural breaks in the series. The plot below suggests that there is a structural break at point number 43, i.e. in the year 1988.In test cricket,this year saw the decline of the mighty team which had dominated the test cricket for ten years.Other teams had become more competitive playing more confidently.For ex- ample, managed to level a three-match series 1-1 against the Windies.The proportion of matches drawn that year dropped significantly to 33% compared to 64% in the previous year.This sharp drop might have resulted in increased competitiveness of the game.Another factor that might have pushed the volatility down is the fact that one-day cricket was getting mature with increased competitiveness and new teams like and performing above par.

9 F-Statistics for Structural Break in Volatility

5 Analysis of Proportion of Matches Drawn Series

Though the popularity of Test cricket might have declined according to some, there is no doubting the fact that more and more games end with a result.We analyse the annual proportion of matches drawn and test for structural breaks. The time plot depicts the percentage of matches that ended without any result for each year.Proportion of drawn matches increased andhovered around 0.44 till the nineteen-eighties. There is a downward trend ever since. The proportion plummeted to 0.24 during the current decade.It seems like a stationary series.

10 ACF and PACF Plots for the Proportion Series

The ACF and PACF plots of the series suggest that the proportion series might be a stationary one.The possible competing models are AR(4),MA(4),ARMA(4,4). Theres no trend in the series. We deduced this by regressing the variable on time. The result is summarized below. The coefficient on the time variable is insignificant at 95% confidence interval.

11 Independent Variable Estimate Standard Error p-value Intercept* 43.72443 3.56057 2e-16 Time -0.17156 0.09674 0.0812 We confirm the absence of any trend in the data by detrending the series and plotting the ACFs and PACFs for the detrended series. There is no change in the behavior of ACFs and PACFs.Further,when we detrended the series,we got an MA(1)series which happens only when the raw series is stationary.Hence, there is no trend whatsoever in the series.

To check for ARCH and GARCH effects we plot the ACFs and PACFs for the residual of the MA(1) model and the residual squared terms.

12 As both the ACF and PACF do not point to any trends in the error term the errors seem to be homoscedastic.Based on the SBC and AIC values, MA (0,0) best describes the first differenced proportion series. The result is summarized below:

Zt = 38.2345 + t (7) where  ∼ N(0, 198.5) (8)

To test for structural breaks in the model , i.e. for the mean value (intercept) we calculate the F statistics for various points for the above regression and plot it. Insert graphs here. The null hypothesis of no structural break is rejected for the points where the F statistic exceeds the critical value indicated by the red line. The point of structural break is the point corresponding to a peak in the plot above the red line. These peaks occur at point 44 and point 48. Thus, the plot suggests two structural breaks in the years 1989 and 1993.

13 We can link the structural breaks to the following events: • 1989 saw the re-emergence of Australian cricket team as a world-class team under the - taincy of .In 1989,the proportion of matches drawn was extremely high(58%). The average proportion of matches drawn in the eighties was 47%.Post 1989,this proportion has dropped to 30%.

• The arrival of South in the test cricket in the 1992-3 was a major event.Very few(less than 30%) matches in which Proteas played ended in draw.

We regressed the points after 1993 on time and found that there is no significant trend during the time-period and that the proportion has remained static during the period.

6 Conclusion

In this paper,we tried to analyse the effects of various events on Test cricket.Had the shorter ver- sions of game affected the test cricket we would have found a change in the behaviour of run rate and proportion of matches drawn.Time series analysis suggests that there has been no sudden change in the movement of run-rate that can be attributed to the events that seemed to have metamorphosed the game. The average run rate has increased with the time without any of the events which we thought might have altered the way run-rate series has behaved.The fact that there is a trend in the time series of the run rate negates the possibility of any sudden changes brought about by the introduction of newer forms of the game in the average run rate of test cricket.There has rather been a gradual rise in the average run rate.The factors which have caused the average run rate in test cricket to rise are pure innovations in the game-the way it is played,the kind of pitches that are used,the technological changes that have occurred and increase in competitiveness in the game. From our analysis,the volatility in the run rate has remained quiescent.The reason for this is the way Test cricket is being played.The run rates rarely jump from band of 2-5 runs per over and most teams prefer a strategy of defensive although in recent times,the trend has changed.For most part of the time period of our analysis,teams stuck to a conventional strategy leading to a low variance in run-rate. The most interesting part of our analysis was to analyse the proportion of matches drawn se- ries.This variable is a measure of competitiveness of the game.The structural breaks signal that

14 there must have been some changes in the game that caused the number of drawn matches to go down significantly.We believe that it’s not only the increase in competitiveness but increase in number of Test cricket playing nations that has contributed significantly to the drop in the proportion of drawn matches. In recent times,many commentators have pointed to the advent of ODIs and emergence of T20s as major factors affecting the quality of Test cricket being played.The statistical analysis we did suggest that inherent inclination towards better quality seems to have driven the changes in Test cricket rather than newer forms of game although these forms of the game have been responsible for some innovations in the game which are essential for these formats.Structural breaks at various points in the series we have considered vouchsafe for the theory that an interplay of various factors has been responsible for the things as they are.

7 Appendix

Chronological Order of Events and their Statistical Significance

Year Event Statistical Effect

1963 • A limited overs tournament called No Structural breaks Gillette Cup was played in 1963 in England.

1971 • Advent of ODI in 1971 is considered to No Structural breaks be a turning point in the history of the game

1977 • World Series Cricket introduced by No Structural breaks Kerry Packer

Late 1980’s • Decline of the mighty Caribbean team Structural Break in 1988 in the Volatility of Run Rate Series

Late 1980’s‐ • Re-emergence of Australian cricket Structural Break in 1989 in the Early 1990’s team. Proportion of Matches Drawn Series

1992-93 • Arrival of which hitherto Structural Break in 1993 in the was banned from Test Cricket due to Proportion of Matches Drawn apartheid. Series • Debut of Zimbabwe in test cricket

2003 • Twenty 20 format introduced No Structural breaks

Following tables show the results for the Box-Jenkins Methodology.

15 16 17 18 19 20 21