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Discuss about the Financial Econometrics. In this report, we will examine how the effects on stock Apple Inc. is influenced by market returns and returns on the information technology sector. 

Overview of Apple Inc.

Apple Inc. is a marketing unit of an American multinational headquartered in Cupertino, California, which develops, develops and sells consumer electronics, software, and online services. Apple is founded in April 1976 by Steve Jobs, Steve Wozniak, and Ronald Wayne to develop and sell the Apple Wozniak computer I. It is recorded in January 1977 under the name Apple Computer, Inc., and the sales of its computers, including Apple II, have increased rapidly (ENGELBERG, and Gao, 2011; Adjasi, Harvey, and Agyapong, 2008).

Stock trading is an economic activity that requires a high degree of precision in its application. Many methods are used in stock trading to maximize returns. Of the many methods used, there are sometimes methods that do not offer the maximum benefit or even the disadvantages. Trading in shares is bound to the country where the company's shares are located. Technical indicators relating to the stock movement curve. The most important part of the formation of stock movements consists of 5 price components: open, high, low, final and transaction volumes.

In this report, we will examine how the effects on stock Apple Inc. is influenced by market returns and returns on the information technology sector. In addition in this article, we consider the correlation between the Apple Inc. and the Standard and Poor 500 (S&P 500) stocks and information technology sector S&P 500 stock indicators. Our research is conducted at four different levels, from the descriptive analysis of daily returns of Apple stock to the impact of S&P indicators’ daily returns on the daily returns of Apple Inc. within a time period of 26th October 2016 to 26th October 2018.

We find that the movement of share prices of Apple Inc. reflected the change in trend starting from the second quarter of 2018. The S&P500 indicators, when compared with the Apple stock price were found to have marked the difference in daily prices. At the stock market level, we find that the daily returns of the S&P500 composite are significantly correlated with the daily return of the Apple prices (Mao, Counts, and Bollen, 2011) 

In a final experiment, we investigate whether a new variable, S&P500 information sector index will have a significant impact we also used a linear regression model to predict market returns for Apple Inc. that use the S&P data as an exogenous input. Specifically, we identify couples in each level and we expect the stock indicators according to the model. Our findings show that S&P indices are useful in predicting the price of Apple shares, and the impact of any structural break in the OLS regression model.

Stock Trading and Technical Indicators

Task -3: Figure 1 shows that the ever-increasing share trend of Apple Inc. and the comparative analysis with S&P500 composite and S&P500 information technology sector share prices is done in the second graph. The normalized values of daily share values have been plotted in the comparative analysis, whereas Apple Inc. daily share values have used in the raw form in the first graph. From the first quarter of 2018, the variation in the standardized curves is noted, where the Apple Inc stock is observed to take an upswing instead of downtrend of S&P Indices. People were probably buying more I-phones, I-pads, or other Apple products with a downward trend of the market.

Task -4: From figure 2 the daily returns are assessed for a two year period of 2016 October to 2018 October. The consistent fluctuation in S&P500 composite is comprehensible, but the rapid variation in daily return in the first quarter of 2018 is an observation to look for. The variation in S&P 500 for information technology shares is also evident in the first and second quarter of 2018. This trend is also clearly visible in

The summary statistics in Table 1 indicates that the average price of Apple Inc. shares (M = $ 38895.82, SD = $ 7480.35) is way ahead of the share price of S&P 500 information technology shares (M = $ 1274.66, SD = $ 202.36). Composite S&P 500 index (M = $4970.89, SD = $ 464.03) is also nowhere near the Apple share prices. Normality of all the three share values is noted from the Jarque-Bera test for normality at 5% level of significance. The descriptive summary of share values reflects the popularity of Apple Inc. shares among investors on the rallying of the stock for the last two years.

The summary statistics in Table 2 indicates that the average daily return on Apple Inc. (M = 0.13%, SD =1.27%) the stock is way ahead of the share price of S&P 500 information technology shares (M =0.09%, SD =1.01%). Composite S&P 500 index (M = 0.05%, SD = 0.69%) is also nowhere near the Apple share prices. Normality of all the three share values is noted from the Jarque-Bera test for normality at 5% level of significance. The descriptive summary of share values reflects that the daily returns for the last two years on Apple Inc. shares have been very encouraging, considering the fact that the trade happens at such high daily stock prices. The market volatility of 1.27% is well understood at that level of share price, where the minimum daily return is observed to be as low as - 4.63% and the highest return as 6.10%. The popularity of Apple Inc. shares among investors is also evident due to that fact that the composite S&P 500 and information technology sector S&P 500 were able to provide daily returns maximum up to 2.72% and 4.03% respectively.

Influence of Market and Sector Returns on Apple Inc. Stock

From figure 3 the relative nature of the daily return normalized curves is apparent, especially the joint fluctuations from the first quarter of 2018. After the first quarter of 2018, high fluctuations and market volatility in the American stock market are noted. Interestingly, the daily returns on Apple Inc. shares also varied with the market volatility in S&P. Moreover, a period can be identified in the third quarter of 2018, where daily returns were comparatively higher than market returns. At the last quarter of 2018, Apple stocks are noted to provide better returns compared to highly volatile S&P 500 stocks,

From the Box plot of the three stocks of the research, we can identify the outlier values. The outliers can be classified into two categories. The red market outlier values are the nearest outliers and the black marked are the outermost outliers, as identified by the E-views environment. It can also be easily interpreted that the spread of the daily return on Apple Inc. stocks is higher than that of the other two indices. It implies that the middle 50% daily market returns were spread out between somewhere between -0.02% and 0.02%. The lower spread of S&P 500 stocks reflected the constrictions due to the overall impact of other stocks.

Task -5: We run two simple regression models of daily stock returns of Apple Inc. on market returns for the S&P 500 composite index, and daily market returns of the S&P 500 IT sector index. The S&P 500 market indices were considered as the predictors and the output variable is taken as the daily market returns on Apple Inc. shares. The normality of the predictors is earlier noted in descriptive summary by JB test.

Table 3 contains the results of the OLS regression model of daily stock returns of Apple Inc. on market returns for the S&P 500 composite index. The coefficient of determination (R-square) indicated that the S&P 500 composite index is able to explain 37.89% variation in Apple Inc. daily market returns. The predictor is found to be statistically significant (t= 17.81, p < 0.01) at 1% level of significance. The intercept is found to be almost zero, and the t-value (t = 1.73, p = 0.085) also justified our initial observation. The ANOVA model is statistically and highly significant (F = 317.22, p < 0.01) at 1% level of significance.  The estimated equation is

The post estimation of the regression model is conducted for the test of linearity of the coefficient and intercept or constant of the model. Wald test is used for the purpose, and it is noted that the slope coefficient is statistically significant for the linear regression model. Whereas, the intercept or constant term is found to be in accordance with the null hypothesis assuming that the value of the term is zero at 5% level of significance. For one-sided t-test for the slope coefficient equal to one. The results have been provided in Table 5, where t-value (t =2.085, p = 0.188) indicated that the null hypothesis assuming the coefficient equal to one failed to get rejected at 5% level of significance. Hence, it can be statistically inferred that the slope coefficient is almost equal to one, implying a 45-degree angle for the regression line.

Results and Analysis

The normality of the residuals so of the regression model is plotted in a histogram and the post-estimation diagnostics revealed that there is consistency regarding the matter, where the statistical confirmation is due to the value of JB test (JB = 1143.26, p < 0.01) at 1% level.

No multi co-linearity and Heteroskedasticity problem are identified in the model. The Variance Inflation Factor (VIF) is confined to one, and the Breusch-Pagan-Godfrey test revealed that Heteroskedasticity quandary does not exist in this model. 

Table 6 contains the results of the OLS regression model of daily stock returns of Apple Inc. on market returns for the S&P500 information technology stocks. The coefficient of determination (R-square) indicated that S&P500 IT is able to explain 59.06% variation in Apple Inc. daily market returns. The predictor is found to be statistically significant (t= 27.39, p < 0.01) at 1% level of significance. The intercept is found to be almost zero, and the t-value (t = 1.42, p = 0.155) also justified our initial observation. The ANOVA model is statistically and highly significant (F = 750.21, p < 0.01) at 1% level of significance.  The estimated equation is.

Task -6: We extend the models of the previous section to a multiple linear regression model with S&P500 composite and S&P500 IT sector as predictors. The dependent or outcome variable is considered as the daily returns of Apple Inc. stock. An OLS regression model is used to estimate the market returns of Apple shares. The normality assumption of the model is satisfied earlier, and the post-estimation diagnostics and tests are also included in the present article.

Table 7 contains the results of OLS multiple regression model of daily stock returns of Apple Inc. on market returns for the S&P500 IT and S&P500 composite stocks. The coefficient of determination (R-square) indicated that S&P500 predictors are able to explain 60.74% variation in Apple Inc. daily market returns. The S&P500 composite (t= -4.72, p < 0.01) and S&P500 IT (t= 17.38, p < 0.01) predictors are found to be statistically significant at 1% level of significance. The intercept is found to be almost zero, and the t-value (t = 1.44, p = 0.148) also defended our initial observation. The ANOVA model is statistically and highly significant (F = 401.61, p < 0.01) at 1% level of significance.  The estimated equation is.

The interaction of S&P500 composite and S&P500 IT inflicted negative slope coefficient for S&P500 composite on Apple Inc. daily returns. In the simple regression model, the direct impact is observed, but no effect of sector funds is taken into consideration. The interaction of S&P500 IT and S&P500 composite is highly significant with a positive Pearson’s correlation of 0.88. This high positive interaction between the predictors is the foremost reason for the negative slope for the S&P500 composite in the current regression model.

OLS Regression Models

The post estimation of the regression model is conducted by the Wald test, and it is noted that the slope coefficients are statistically significant for the linear regression model. Whereas, the intercept or constant term is found to be in accordance with the null hypothesis assuming that the value of the term is zero at 5% level of significance. The t-tests implied that the slope of the S&P500 composite is (t = -4.72, p < 0.01) significantly different from zero, and the relation with the outcome variable is linear in nature. The slope coefficient of S&P IT is also statistically significant (t = 17.38, p < 0.01), and reflected a significant linear association with the outcome variable. The intercept value (t = 1.44, p = 0.148) failed to reject the null hypothesis of its significance in the regression equation.

The normality of the residuals so of the multiple regression model is plotted in a histogram and the post-estimation diagnostics revealed that there is consistency regarding the matter, where the statistical confirmation is due to the value of JB test (JB = 1305.19, p < 0.01) at 1% level.

Multi co-linearity is observed between the two independent predictors of the model. The VIF value = 4.46 for S&P 500 IT and VIF = 4.46 for S&P500 composite indicated occurrence of multi co-linearity due to high positive correlation among the predictors. Heteroskedasticity issue is not present in the model, where the confirmatory analysis was done by the Breusch-Pagan-Godfrey test, which revealed that Heteroskedasticity issue does not exist in this model. The multiple-regression model instituted that both the S&P market returns were imperative predictors of Apple Inc. daily returns on stock prices.

Task -7: Recent research suggests that the joint consideration of the slope coefficients of the multiple regression models, equal to zero is tested by Wald F-test at 5% level of significance. The null hypothesis was constructed assuming that both the slope coefficients are zero and was tested against the alternate hypothesis that there is at least one coefficient not equal to zero, at a 5% level of significance. The F-value (F = 401.61, p <0.01) implied that the null hypothesis is rejected at 5% level of significance, accepting the fact that slope coefficient is not equal to zero. The linearity nature of the variables in the multiple regression models is established.

Task -8: We modified the present multiple regression model by introducing a dummy index for the time period of the two year return period of the data. A structural break was investigated for the Chow test with the structural test. Dickey-Fuller autoregressive model for unit root was used in E-views environment to search for a particular date as a breakpoint, from which market volatility was irregular and haphazard in nature (Narayan, and Popp, 2010). The Dickey-Fuller identified a statistically significant break at 20th April 2018 (t = -22.64, p < 0.01).

Multiple Linear Regression Model

After we have determined that the predictors correlate with stock market indicators, we want to know if stock market indicators can be accurately predicted and how they can be predicted using Apple data. To answer this question, we apply multiple-linear regression with an exogenous input model on our Apple Inc. daily returns on predictors and stock market indicators. At the composite level, we find that the best predictive model for the predictions can explain 37.89% variation in daily returns of Apple stock, indicating that the inclusion of S&P IT data is useful for making more accurate predictive models. In particular, the structural break in the model assigns 1 to the daily returns of the stocks after 20th April 2018 and zero to the daily returns before that date (Kejriwal, and Perron, 2010).

From Table 14 and Table 15 it was noted down that the Chow test scrutinized the breaks at two break points. The estimated regression equations for the two models are considered as

The null hypotheses for the test were H0: , H0: , and H0: , the residual sums for the constrained and unconstrained models were considered to find the value of the test statistic with F(K, n-2*k-2) (Arai, and Kurozumi, 2007).

For the breakpoint on 20th April 2018, no significance difference in regression models was observed (F (3, 516) = 1.704, p = 0.165)) at 5% level of significance.  The Chow test is again performed for a break at 30th April 2018. This time the significance value improved to reject the null hypothesis at a 10% level of significance. With 90% confidence, it can be said that the predicted daily returns of Apple Inc. and its estimated equation after 30th April 2018 differed just significantly from that of the estimated regression equation till 30th April 2018. But, at 5% the null hypothesis failed to get rejected, concluding that there was a statistically significant difference in estimation model pre and post the structural breaks (Aue, and Horváth, 2013).

Conclusion

We have examined whether the daily returns of Apple Inc. listed in S&P500 correlates with multiple S&P 500 stock and IT S&P 500 indexes. This scrutiny happened at three different levels, from the composite stock market, individual sector returns to multiple regression model with structural breaks. AT first, the simple linear model was constructed to evaluate the impact of S&P composite on the daily returns. Secondly, the combined impact of S&P 500 composite and IT sector stocks was investigated. Due to the multi co-linearity issue, the composite daily return predictor was noted to negatively impact the Apple stock returns. Further investigation yielded that no structural difference in the regression models existed for a break near the first quarter of 2018. The diversity in market volatility from other time periods was identified in the descriptive analysis section. At 30th April 2018, a significant break was estimated, and we found that the break was just not significant at 5% level of significance.

Conclusion

Our results show that the daily market return correlates with stock market indicators. It also appears that data from Apple Inc. can be useful in predicting stock markets. As a future work, we will follow the following directions (Makrehchi, Shah, and Liao, 2013). In the first place, we search for more heterogeneous data to collect more relevant trends on a daily basis. Secondly, we collect long-term data and evaluate the results. Thirdly, we will combine the results of Apple Inc. and that sentiment in the purchase of Apple products. Finally, we check whether the predictive results can help a trader take business decisions. We can also propose a new method based on emotion. In Apple stocks we feel that to predict future actions in market movement, the sentimental purchase pattern is required for proper forecasting. In practice, we can estimate with high accuracy and effective classification for evaluating and using sentiment information on building an effective trading strategy (Jin, Gallagher, Cao, Luo, and Han, 2010). The strategy will be able to beat predictions by S&P500 indices only. We used a linear regression with an exogenous input model to predict Apple stock indicators by using S&P data. The forecasting model to validate the regression model framed in this study is required in future research work (Hu, 2014; Zhong, and Enke, 2017).

The technique with the specification in the model has been described in this article. It requires rather strict assumptions: if linearity is rejected, the alternative can only be for the star model. On the other hand, the technique it's easy to use and show the examples in the article. It works pretty well, already in relatively small samples. Keep in mind that the same method also works if the only one that is not linear the model is a tar model with a threshold because the F-tests also have a force against these borderline cases. However, in many programs are possible for the transition model to run smoothly. As macroeconomic the limits are usually the result of decisions taken affects the decisions or changes (De Castro, Franco, Pinto, and Seo, 2009; Nassirtoussi, Aghabozorgi, Wah, and Ngo, 2014).

References

Adjasi, C., Harvey, S. and Agyapong, D., 2008. Effect of exchange rate volatility on the Ghana Stock Exchange.

Arai, Y. and Kurozumi, E., 2007. Testing for the null hypothesis of cointegration with a structural break. Econometric Reviews, 26(6), pp.705-739.

Aue, A. and Horváth, L., 2013. Structural breaks in time series. Journal of Time Series Analysis, 34(1), pp.1-16.

De Castro, R.C.F., Franco, L.G., Pinto, A.G. and Seo, C.E., International Business Machines Corp, 2009. System and method for automated stock market operation. U.S. Patent Application 12/235,830.

ENGELBERG, J. and Gao, P., 2011. In search of attention. The Journal of Finance, 66(5), pp.1461-1499.

Hu, J., 2014. Does option trading convey stock price information?. Journal of Financial Economics, 111(3), pp.625-645.

Jin, X., Gallagher, A., Cao, L., Luo, J. and Han, J., 2010, October. The wisdom of social multimedia: using flickr for prediction and forecast. In Proceedings of the 18th ACM international conference on Multimedia (pp. 1235-1244). ACM.

Kejriwal, M. and Perron, P., 2010. Testing for multiple structural changes in cointegrated regression models. Journal of Business & Economic Statistics, 28(4), pp.503-522.

Makrehchi, M., Shah, S. and Liao, W., 2013, November. Stock prediction using event-based sentiment analysis. In Proceedings of the 2013 IEEE/WIC/ACM International Joint Conferences on Web Intelligence (WI) and Intelligent Agent Technologies (IAT)-Volume 01 (pp. 337-342). IEEE Computer Society.

Mao, H., Counts, S. and Bollen, J., 2011. Predicting financial markets: Comparing survey, news, twitter and search engine data. arXiv preprint arXiv:1112.1051.

Mao, Y., Wei, W., Wang, B. and Liu, B., 2012, August. Correlating S&P 500 stocks with Twitter data. In Proceedings of the first ACM international workshop on hot topics on interdisciplinary social networks research (pp. 69-72). ACM.

Narayan, P.K. and Popp, S., 2010. A new unit root test with two structural breaks in level and slope at unknown time. Journal of Applied Statistics, 37(9), pp.1425-1438.

Nassirtoussi, A.K., Aghabozorgi, S., Wah, T.Y. and Ngo, D.C.L., 2014. Text mining for market prediction: A systematic review. Expert Systems with Applications, 41(16), pp.7653-7670.

Zhong, X. and Enke, D., 2017. Forecasting daily stock market return using dimensionality reduction. Expert Systems with Applications, 67, pp.126-139.

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