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26-March-2006
Does the First Quarter Predict the Rest of the Year? By Victor Niederhoffer

The following borders on the useful, except for the retrospection, changing cycles and multiple hypotheses that characterize all such studies. However, for the good of our dedicated weekend readers, I put it up immediately, before the Minister of Non-Predictive Studies returns from his weekly weekend tournament outings -- Vic

I queried recently if a good first quarter of the year had any predictive significance for the remainder of the year. The question arose from my observation that this year, first quarter is up 5% to date, and 40% of years have seen 20% or more rises before dividends.

Doc Castaldo has kindly enumerated the data for the first quarter's performance and the subsequent three quarter's performance, (please excuse all those numbers to the right of the decimal). One notes, as he did, that when the first quarter was up the average change over the next three quarters was 9%, whereas when the first quarter was down, the average change over the next three quarters was up a mere 1%.

Using simulation, a difference as large as this is about one in a hundred to have arisen by chance. One notes also that all but one of the bad final nine months of the year were preceded by down first quarters, the one exception being 1987 when a 20% first quarter was followed by a down move of 15% in the last three quarters. One also notes that fourteen, or 25%, of the first quarters were up between 4 and 6%. Such a clustering appears to be non-random, however, more importantly one notes that none of these quarters was followed by a decline the rest of the year, with the average change for the rest of the year being some 11%, with 10 of the last three quarters clustering around a gain of 10%, plus or minus 5%. Again this appears non-random. Despite this, within the 32 first quarters that were up, there does not appear to be a linear association between the first quarter of the years performance and the subsequent three quarters.

One queries whether such results are numerology or a benchmark? One also queries whether such results might be extended to individual stocks? Also, whether classifying independent variables such as the above by clusters of their magnitudes, a la the analysis of the intervals of skips between consecutive notes in my report on musical compositions, (Mozart was much more regular than Scriabin), or conditional on classes of the independent variable, a la my report on the statistical analysis on regression forecasting, might be more appropriate?

One also queries whether the other quarters show similar relations and whether such relations could have been discovered on a prospective basis, rather than by the retrospective approach above? Finally, whether there is a general relationship between predictive patterns based on retrospective seasonal analysis and the subsequent out of sample performance of such relationships, (I posit a 90% reversion to the mean)?

The S&P returns since 1950 (terminating at month end).

We can define a dummy variable to be 1 if the 1st quarter is up and 0 otherwise. A regression with the Q2Q4 return as dependent and the dummy as independent gives the following results, (courtesy of Alex Castaldo):

Summary Output:

Regression Statistics

```Multiple R        0.310541
R Square          0.096436
Adjusted R Square 0.079703
Standard Error    0.128288
Observations            56```

ANOVA

```                df      SS        MS        F      Significance F
Regression       1   0.094852  0.094852  5.763328    0.019839
Residual        54   0.888727  0.016458
Total           55   0.983579```
```                 Coeffici   Standard   t-Stat     P-value
Intercept        0.014621   0.027351   0.534569   0.595143
Q1 is Positive   0.084269   0.035102   2.400693   0.019839
```

Thanks to Doc, Tex, and Mr. Saur for calculations and/or the approaches alluded to above. One encourages other queries and, with the Minister's permission, other useful answers sparked by the above beginnings.

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