The Influence of Weather Conditions on Rates of Return of Polish Equity Indices

The influence of the weather on human behavior have been featured in many, not only scientific publications. This paper tests the hypothesis that the one-session average rates of return of equity indices (WIG, WIG20, mWIG40 and sWIG80) calculated for the different weather conditions differ in two populations. The atmospheric conditions taken into consideration in this paper are as follows: maximum and minimum daily temperature, sunny hours, rainfall, maximum and average wind velocity, atmospheric pressure, snow depth, sun energy ultraviolet radiation index. In the analyzed period, the impact on the daily rates of return was observed in the case of the following weather conditions: maximum daily temperature, sunny hours, rainfall, maximum wind velocity and atmospheric pressure. The other analyzed weather conditions such as average wind velocity, minimum daily temperature, snow depth, sun energy ultraviolet radiation index, turned out to be irrelevant. Thus, the influence of some weather condition on registered rates of return on the Polish equity markets has been proved.


Introduction to the Problem -Weather Influence on Investors Decisions
Seasonality of equity market has a long history although the academic research has been dominated by efficient market theory introduced by Fama (1970Fama ( , 1991)).Hirshleifer (2001), Hirshleifer and Shumway (2003) proved that psychologists documented correlation between sunshine and human behavior.Sunshine was lined to tipping (Rind, 1996) and lack of sunshine to depression (Eagles, 1994).According to Tietjen and Kripke (1994) the number of sunny hours influenced the number of committed suicides.People feel good when the sun shines and are more optimistic, consequently they are prone to open a long position in stocks what leads to the higher level of stock prices -therefore the weather influence on investors is very often discussed in papers dedicated to behavioral finance (Akhtari, 2011).
The main purpose of this paper is to verify the null hypothesis that the difference of one-session average rate of return calculated for the session, when the particular atmospheric conditions were present, and the one-session average rate of return, calculated for all sessions, when the particular atmospheric conditions were absent, is equal zero.The weather conditions analyzed in this research are as follows: maximum and minimum daily temperature, sunny hours, sun energy, rainfalls, maximum and average wind velocity, atmospheric pressure, snow depth and ultraviolet radiation.This is the first research regarding weather conditions influence on rates of return on the polish equity market.

Literature Review
The scientific literature dedicated to the weather influence on the capital market is not robust.According to Roll (1992) "weather data should be used in assessing the information processing ability of financial markets" and "weather is a genuinely exogenous economic factor".Saunders (1993) research was one of the first studies dedicated to the effects of cloud cover on stock returns.The author concentrated on daily returns of the Dow Jones Industrial Average in the time span of 1927-1989, applying the "percentage of cloud cover from sunrise to sunset" according to the New York weather station closest to Wall Street.Saunders (1993) found that stock returns are lower during the session of 100% cloud cover than on days when cloud cover was 20% or less and discovered that the weather effect impact on rates of return was higher in the period of 1982-1989 then in the previous analyzed period 1962-1982.To the similar conclusion came Akthari (2011) who based his research on the period of 1948-2010 in USA and noticed that the weather effect was stronger in some periods.Hirshleifer and Shumway (2003) based on the data from 26 stock exchanges for the period of 1982-1997, proved that sunny weather was associated with an upbeat mood of investors and found the strong correlation between stock rates of return and sunshine.The authors discovered that stock returns tended to be higher on sunny days but other weather conditions such as rain and snow were unrelated to returns.
Taking into consideration seasonal time-variation of risk premia in stock returns, Kamstra et al. (2001), referring to yearly daylight fluctuations, introduced the Seasonal Affective Disorder (SAD) and found statistical significance relationship between stock returns and SAD.According to the authors, the lack of sunlight induced investors depression, what increased their risk aversion and affected stock valuation.Keef and Roush (2007) based on 26 international stock exchanges showed the relation between the sunshine effect, per capita GDP and latitude.They found no effect in the stock exchanges situated near the equator and a big influence on northern exchanges.Trombley (1997) suggested that the relation between weather conditions and stock return is not as obvious as Saunders (1993) claimed, and proved that the weather influence on rates of return appeared in only one of the three periods examined, and the effect was limited to only a few months of the year, while ceasing in the others.According to Trombley (1997) the strongest weather effect impact on US stock prices, was observed in the period of 1990-1997.Loughran and Schulz (2004) found the little evidence conforming that cloudy weather in the city in which analyzed company was based, affected its returns.Pardo and Valor (2002) investigated the relation between weather and rates of return on the Spanish market (Madrid Stock Exchange) and analysing daily closing prices, they proved that in the examined period there was no impact of sunny hours and humidity levels on equity rates of return.Another negative evidence was presented by Kramer and Runde (1997), who whilst investigating weather effect on Frankfurt Stock Exchange in the period of , discovered that weather conditions were irrelevant to the short-term stock returns.According to them the positive or negative results relating weather influence on market returns result from different statistical methods applied by researchers.

Method
According to the adopted methodology, the survey covers two populations of returns, characterized by normal distributions.On the basis of two independent populations of rate of returns, which sizes are equal n 1 and n 2 respectively, the hypotheses H 0 and H 1 should be tested with the use of statistics z (Osinska 2006, pp. 43-44) where: 1 ̅ -average rate of return in the first population,  2 ̅ -average rate of return in the second population,  1 -number of rates of return in the first population,  2 -number of rates of return in the second population,  1 2 -variance of rates of returns in the first population,  2 2 -variance of rates of returns in the second population.
The Formula 1 can be used in the case of normally distributed populations, when the populations variances are unknown but assumed equal.The number of degrees of freedom is equal to: (1) =  1 +  2 − 2.
Because the population variances are unknown, it might occur that the populations variances are unequal.In such case we can use the Formula 1 to calculate the z statistics, but the number of degrees should be modified according to the following formula (Defusco et al., 2001, p. 335): (2) In case of two populations, both with equal or unequal variances, the null hypothesis H 0 and alternative hypothesis H 1 regarding equality of rates of return in two populations, can be formulated as follows: In particular: for the analysis of the daily rates of return, if  1 ̅ is the daily average rate of return calculated for sessions when the particular weather condition was observed (the first population), then  2 ̅ is the daily average rate of return computed for all other session, when the particular weather condition was not observed (the second population).
In all analyzed cases, the p-values will be calculated with the assumption that the populations variances are unknown, but: a) population variances are assumed equalp-value(1), b) population variances are assumed unequal -p-value(2).
If the p-value is less than or equal to 0,05; then the hypothesis H 0 is rejected in favor of the hypothesis H 1 .Otherwise, there is no reason to reject hypothesis H 0 .
As the last part of the calculation will be carried out F-test using the F-statistics (so called Fisher-Snadecor statistics) for equality of variances of two population rates of return, where  =   2   2 , with the condition that:   2 >   2 and that ,  = 1,2; and the degrees of freedom are equal: for variance in the numerator of F,  for variance in the denominator of F.
If F-test (computed for α=0,05) is lower than F-statistics, there is no reason to reject the null hypothesis, which can be formulated as follows: The alternative hypothesis may be defined by the ensuing equation: (5) In the case, when there is no reason to reject the null hypothesis concerning equality of variances of two observed returns, the p-value(1) should be compared with the critical value 0,05; otherwise the p-value(2) will be usedthat explains the reason of applying p-value in the following part of the paper.
All analyzed weather conditions refer to Warsaw and were measured in the period of 01.01.2008 -30.06.2015.Some of the weather conditions data was downloaded from the website: http://www.andretti.plon which are revealed details regarding ultraviolet radiation index.The statistical hypothesis were tested for the following indices: WIG (the broad market index), WIG20 (blue chips index), mWIG40 (medium capitalization companies index) and sWIG80 (small capitalization companies index).

Analysis of the Maximum Daily Temperature Influence on Rates of Return
The results of testing the null hypothesis permit to draw the following conclusions: 1) The null hypothesis regarding equality of variances of daily average rates of return in two populations, was rejected (for α=5%) in the following cases: 2) The null hypothesis regarding equality of two average rates of return in two populations was rejected in the following cases (p-value shown in parenthesis): a) WIG index -category 10-15 (0,0176).
In all analyzed cases, the p-value was higher than 0,05; and therefore there was no reason to reject the null hypothesis regarding equality of averages in two group of populations.

Analysis of the Number of Sunny Hours Influence on Rates of Return
The results of testing the null hypothesis permit to draw the following conclusions: 1) The null hypothesis regarding equality of variances of daily average rates of return in two populations, was rejected (for α=5%) in the following cases: a) WIG index -categories (below 1, 1-2, 6-7, 7-8 and 10-11).
2) The null hypothesis regarding equality of two average rates of return in two populations was rejected in the following cases (p-value shown in parenthesis): a) WIG index -category 5-6 (0,0335).For the category 9-10, the p-value was equal 0,0678.
The one session average rates of return for each of the analyzed indices broken down into the daily number of sunny hours are presented on the figure 1.The highest number of daily average rates of return highest than zero was observed in the case of mWIG40 ( 9) and the lowest for WIG20 (6), whilst for two other indices this number mounted to 8.
Figure 1.The one session average rates of return for 4 analyzed indices broken down into the daily number of sunny hours Source: own calculations.

Analysis of the Daily Rainfall Influence on Rates of Return
The results of testing the null hypothesis permit to draw the following conclusions: 1) The null hypothesis regarding equality of variances of daily average rates of return in two populations, was rejected (for α=5%) in the following cases: a) WIG index -category 6-10.
2) The null hypothesis regarding equality of two average rates of return in two populations was rejected in the following cases (p-value shown in parenthesis): a) WIG -category 20-25 (0,0197).
The one session average rates of return for each of the analyzed indices broken down into the daily amount of rainfall are presented on the figure 2. The highest number of daily average rates of return highest than zero was observed in the case of mWIG40 (8) whilst for three remaining indices it was equal to 7.
- WIG WIG20 mWIG40 sWIG80 Figure 2. The one session average rates of return for 4 analyzed indices broken down into daily amount of rainfall Source: own calculations.

Analysis of the Average Wind Velocity Influence on Rates of Return
The null hypothesis regarding equality of variances of daily average rates of return in two populations, was rejected (for α=5%) in the following cases: a) WIG index -categories: less than 1 and 1-5,5.
In all analyzed cases, the p-value was higher than 0,05; and therefore there was no reason to reject the null hypothesis regarding equality of averages in two group of populations.

Analysis of the Maximum Wind Velocity Influence on Rates of Return
The results of testing the null hypothesis permit to draw the following conclusions: 1) The null hypothesis regarding equality of variances of daily average rates of return in two populations, was rejected (for α=5%) in the following cases: a) WIG index -categories less than 1; 1-5,5; and 5,5-11.
2) The null hypothesis regarding equality of two average rates of return in two populations was rejected in the case of mWIG40 in category: 1-5,5, for which the p-value was equal to 0,0277.For the categories 5,5-11 and 28-28 the p-value were equal to 0,0529 and 0,0890, respectively.

Analysis of the Snow Depth Influence on Rates of Return
The null hypothesis regarding equality of variances of daily average rates of return in two populations, was rejected (for α=5%) for all of analyzed indices in 13 categories.
In all analyzed cases, the p-value was higher than 0,05; and therefore there was no reason to reject the null hypothesis regarding equality of averages in two group of populations.