Applied statistics for economics - Essay Example

? Due: Stock market prediction has been of major interest to the statisticians and econometricians for quite some time. Having an accurate idea and knowledge relating the future performance helps traders and farmers to invest in the most viable business opportunities thus maximizing the profits. Analysis of time series data and forecasting has been used in many fields and most commonly in the stock market prediction using the past data. INTRODUCTION Time series analysis is a form of statistical data analysis on a series of sequential data points that are usually measured at uniform time intervals over a period of time. A time series can be said to collection of data yt (t=1,2,…,T), with the interval between yt and yt+1 being fixed and constant. Time series analysis is the estimation of difference equations containing stochastic (error) terms (Enders 2010). Time series forecasting takes the analysis from the time series data and tries to predict what the data may be in the near future, based on what it has been in the past. This is especially important in the field of stock market investment, as traders want to make the right moves at the right times to maximize financial profit. But because there are many factors influencing the fluctuation of the stock market, creating an accurate forecast based on the analysis alone is difficult. Therefore, many approaches and models have been developed to utilize the time series analysis and provide an accurate prediction of what is to come in the stock market. Standard & Poor 500 indices are designed to reflect the U.S. equity markets and, through the markets, the U.S. economy. The Standard & Poor 500 index focuses on the large-cap sector of the market; however, since it includes a significant portion of the total value of the market, it also represents the market. Companies in the Standard & Poor 500 index are considered leading companies in leading industries. PURPOSE OF THE REPORT The report also examines the impact of exogenous changes in stock prices on the standard & poor 500 index. Specifically, the report investigates whether following a stock price decline, standard & poor 500 indexes disclose information concerning firm value that was withheld prior to the stock price decline because it was “unfavorable” and became favorable at lower stock prices. Consistent with the research predictions, the study found out that standard & poor 500 indexes are more likely to release good news forecasts following larger stock price declines. Moreover, as expected there is no association between the likelihood of releasing good news forecasts and the magnitude of an exogenous stock price increase. Also to be illustrated in this report is the potential of Time-frequency Representation (TFR) techniques, for the analysis of stock markets data and compare their performances. It is for this reason that we seek to analyze the behavior of the stock market price over the period from 2002 to 2007. MAIN OBJECTIVES/BROAD PURPOSES OF THE PROJECT This report aims at investigating the following objectives; i) To investigate whether all the investors are risk averse and measure risk in terms of the standard deviation in a portfolio’s return. ii) To investigate whether all investors have a common time horizon for investment decisions (e.g., a year). iii) To investigate whether all investors have identical subjective estimates of future returns and risks for all securities. iv) To investigate whether there exists a risk-free asset and all investors may borrow or lend an unlimited amount at the risk free nominal rate of interest. v) To investigate whether all securities are completely divisible, there are no transactions costs or differential taxes, and there are no restrictions on short-selling. vi) To investigate whether information is freely and simultaneously available to all investors. We could summarize the above by stating that all individuals are price-takers and that markets are perfectly competitive. OUTCOMES OF THE PROJECT Comparing the mean of dly and that of dlsap, we observe that the end of week mean price for the logarithm of share price for a particular stock (denoted dly) is lower compared to the end of week mean for the logarithm of Standard & Poor 500 index From the data it can be observed that the mean dly is 0.0004162 while that of dlsap is 0.0007498 with stand deviations of .0459125 and .0311088 respectively. The minimum and maximum values of dly are -.2434263 and .2434263 respectively. While that of dlsap, i.e. the minimum and maximum values, are -.1618743 and .1618743 respectively. . summarize dly dlsap Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- dly | 256 .0004162 .0459125 -.2434263 .2434263 dlsap | 256 .0007498 .0311088 -.1618743 .1618743 There is a strong correlation between the dly and dlsap; from the table generated below on the correlations, the correlation coefficient = 0.8849* implying a strong positive correlation between dly and dlsap . pwcorr dly dlsap, star(5) | dly dlsap -------------+------------------ dly | 1.0000 dlsap | 0.8849* 1.0000 To explain the concept of the correlation further, a scatter plot representing the dly and dlysap is plotted and from the pattern of the graph, it can easily be seen that there is a strong positive correlation/relationship between dly and dlsap. For instance, for any unit increase in dly there is a subsequent increase in dlsap likewise for any unit drop in dly there is a proportional drop in dlsap. In order to check on the stationarity or non-stationarity of the data, the time series plot were generated and they are represented below; going as per the pattern of the graph, it can be observed that clearly dly exhibits the property of being stationary. This implies that the mean, the variance and the autocorrelation of the dly data are all constant over time. Testing for Stationarity (Integration Filter); (i) mean: E(Yt) = ? (ii) variance: var(Yt) = E( Yt – ?)2 = ?2 (iii) Covariance: ?k = E[(Yt – ?)(Yt-k – ?)2 Forms of Stationarity: weak, strong (strict), super (Engle, Hendry, & Richard 1983) Just like the above time series for dly, the figure below was generated to check the property of stationary in dlsap and clearly it can be observed that dlsap exhibits the property of being stationary-explained by the pattern of the time series graph. Similarly, the above observation implies that the mean, the variance and the autocorrelation of the dlsap data are all constant over time. The above findings can further be explained basing on the table generated below. In this table, the mean of the variables for the first quarter are compared against the mean of the second quarter. If large variations exist then such a variable exhibits non-stationary property while the converse is true. Variable Sample means of Newts for sample periods Remarks 2002w1-2004w1 2004w1-2007w1 Real y 21.55 22.07 Change in mean differ across the periods-non-stationary Real sap 1130.75 1182.89 Change in mean differ across the periods-non-stationary Change in dly -.00193 0.0013 Change in mean do not differ across the periods-stationary Change in dlsap -.0003 0.0012 Change in mean do not differ across the periods-stationary TESTS OF THE BASIC MODEL The basic equation tested is; The table below gives the output data; logit dly time_dum Iteration 0: log likelihood = -28.44964 Iteration 1: log likelihood = -28.08025 Iteration 2: log likelihood = -28.069096 Iteration 3: log likelihood = -28.069086 Logistic regression Number of obs = 256 LR chi2(1) = 0.76 Prob > chi2 = 0.3830 Log likelihood = -28.069086 Pseudo R2 = 0.0134 ------------------------------------------------------------------------------ dly | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- time_dum | .7411564 .8752183 0.85 0.397 -.97424 2.456553 _cons | 3.417727 .5081305 6.73 0.000 2.421809 4.413644 From the above table, the likelihood ratio chi-square of 0.76 with a p-value of 0.3830 tells us that our model as a whole does not fit significantly better than an empty model (i.e., a model with no predictors). Also we can observe in the table the coefficients, their standard errors, the z-statistic, associated p-values, and the 95% confidence interval of the coefficients.  Time_dum is statistically insignificant to mean that no relationship exists between dly and the time_dum. The other test conducted was based on the equation; logit dlsap time_dum Iteration 0: log likelihood = -16.322048 Iteration 1: log likelihood = -16.139645 Iteration 2: log likelihood = -16.133996 Iteration 3: log likelihood = -16.13399 Logistic regression Number of obs = 256 LR chi2(1) = 0.38 Prob > chi2 = 0.5397 Log likelihood = -16.13399 Pseudo R2 = 0.0115 ------------------------------------------------------------------------------ dlsap | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- time_dum | .732678 1.231183 0.60 0.552 -1.680397 3.145753 _cons | 4.127134 .7127862 5.79 0.000 2.730099 5.52417 Similar to the earlier findings for the dly, the above table gives the logit regression for dlsap with respect to the dummy time; the likelihood ratio chi-square of 0.38 with a p-value of 0.5397>0.05 (significance level) tells us that our model as a whole does not fit significantly better than an empty model (i.e., a model with no predictors). Also we can observe in the table the coefficients, their standard errors, the z-statistic, associated p-values, and the 95% confidence interval of the coefficients.  Time_dum is statistically insignificant to mean that no relationship exists between dly and the time_dum. DICKER-FULLER TEST FOR UNIT ROOT When we conduct the Dicker-Fuller Test and Phillips-Perron Test), we would reject the null hypothesis of a unit root (With both tests; ADF, Phillips-Perron) and conclude that the approval series is stationary. This makes sense because it is hard to imagine a bounded variable (0-100) having an infinitely exploding variance over time. Though however, most scholars have shown that the series does have some persistence as it trends upward or downward, suggesting that a fractionally integrated model might work best (Box-Steffensmeier & De Boef). The Dickey–Fuller test involves fitting the regression model ?yt = ?yt?1 + (constant, time trend) + ut (1) By ordinary least squares (OLS), but serial correlation will present a problem. To account for this, the augmented Dickey–Fuller test’s regression includes lags of the first differences of yt. The Phillips–Perron test involves fitting (1), and the results are used to calculate the test statistics. They estimate not (1) but: yt = ?yt?1 + (constant, time trend) + ut In (1) ut is I(0) and may be heteroskedastic. The PP tests correct for any serial correlation and heteroskedasticity in the errors ut non-parametrically by modifying the Dickey Fuller test statistics. Considering dly, the MacKinnon approximate p-value for Z(t) = 0.0000>0.05 (significance level) implying it highly significant Dickey-Fuller test for unit root Number of obs = 255 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -23.608 -3.460 -2.880 -2.570 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0000 . dfuller dlsap, lags(0) Dickey-Fuller test for unit root Number of obs = 255 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -23.122 -3.460 -2.880 -2.570 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0000 pperron dly Phillips-Perron test for unit root Number of obs = 255 Newey-West lags = 4 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(rho) -294.968 -20.304 -14.000 -11.200 Z(t) -28.336 -3.460 -2.880 -2.570 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0000 . pperron dlsap Phillips-Perron test for unit root Number of obs = 255 Newey-West lags = 4 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(rho) -289.881 -20.304 -14.000 -11.200 Z(t) -27.784 -3.460 -2.880 -2.570 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0000 CONCLUSION This paper presents one major approach, Time-frequency Representation (TFR) that creates a time series analysis and forecast for a stock market. Due to the scope of the paper, the paper only briefly explores the different approaches and how time series data is utilized by different models to achieve forecasting. It is important to note based on the analysis there is no much difference between the stock price and the Standard & Poor 500 indices. REFRENCES 1. Andrews, Donald W. K. (1991), .Asymptotic normality of series estimators for nonparametric and semi parametric regression models,.Econometrica, 59, 307-345. 2. Breusch, T.S. and A.R. Pagan (1979): .The Lagrange multiplier test and its application to model speciation in econometrics,. Review of Economic Studies, 47, 239-253. 3. Chow, G.C. (1960): .Tests of equality between sets of coefficients in two linear regressions,. Econometrica, 28, 591-603. 4. Frisch, Ragnar and F. Waugh (1933): .Partial time regressions as compared with individual trends,.Econometrica, 1, 387-401. 5. Frisch, Ragnar Anton Kittel. Econometrica. Econometric Society, JSTOR, 1943. 6. 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