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Use the files cons income. dta which is stata data file with data on real personal disposable income, RPDI and real personal consumption, rc. The third variable is ‘time’. The data are quarterly from 1947 q1 to 2009 q2. Take the natural logarithms of RPDI and RC, and then first the difference these variables to get the approximate growth rates. So for income: gen LRPDI = ln( RPDI) = D. Lrpdi = d. Lrpdi and similarly for consumption. Estimate a VAR (1) and VAR (2) for the four variables, including two dummy variables (dumy, dumo).

Which is preferred on AIC? From: DL rpdi = ln (RPDI), where RPDI = P1 – PT P1 = personal income and PT = personal taxes Thus, the difference is 0.0197 From the estimation of alternative Univariate method for the differences of DL RDI and DL RC using the observations of income up to 1947, the comments follow the presentation of data estimates. They include autocorrelation of RPDI against Bartletts formula for M. A (q) 95% confidence band. LS against time frame of between 1947 and 2009. And Autocorrelation against Bartletts formula for M. A (q) 95% confidence band.

These variables drawn represent the autocorrelation of the income variable in the country during the period of 1947 and 2009. The representation of the relationship appears in two forms namely definite shape and negative form. For example, a positive autocorrelation indicates persistence in the series of numbers presented. In addition, it means the ability of a series of number or system represented to remain on the same trend from one point of observation to the next view point. Autocorrelation appears complicated in analysing of data.

For example through reducing the number of possibility of independent comments made. Thus, identifying the vital covariance between the series of time, for instance from 1950 to 2010, may appear complicated. However, autocorrelation appears prone to exploitation hence making room for predictions that increase the rate of probability of predicting the future variables or events from the past values. In the analysis of the data provided by the autocorrelation, the tools used for such assessment include time series plot, lagged scatter spot and autocorrelation function. 2) Test for autocorrelation (LM tests) after estimating the VAR (2) and after for VAR (1).

What do these tests suggest? Estimate the cointegrating rank for a second order VAR, including a constant but not a trend. What do you need the cointegrating rank to be? Repeat with a restricted trend (this allows for a trend in the cointegrating relationships). How do the results concerning the cointegrating rank change? Time series plot illustrates both positive and negative forms. Active series have positive meanwhile negative series have a negative mean from their departures following them. Positive autocorrelation indicated in the time series plot appears unusual with long stretches that contain different observations either above the average value or below.

Negative autocorrelation means significant little times of runs. In evaluating the autocorrelation, a horizontal line drawn from the sample mean enables computation of the departures either negative or positive deviations. Thus, in determining the persistence of differences, visual assessment helps in making observations. For example, on observing time series remain constant with number of runs, this indicates persistence. Hence, it may predict other parts in the series. The lagged scatter plot uses simple graph summary of autocorrelation in time series.

For example, random scattering of points in lagged scatters plot results to lack of autocorrelation. Thus, the random series has value of time as independent of the value of other times in the series. For the analysis of lagged scatter plot, alignment in the series from lower left to the upper right indicates a persistence or positive autocorrelation. Analyse the statistical properties of the logs of the series and the differences of the logs of the series: plot the data and look at ACF and PACF.

What features do you notice? Any adjustment from the top left to lower right in time series shows negative autocorrelation. However, the use of lagged scatter plot remains limited to the set of time series due to the mode of presentation in scatter plots, it indicates the same information conveyed in the time plot analysis. In the analysis of autocorrelation of LS plot against Time from Bartletts formula for 95 % confidence band uses time series plot. For example in the beginning of the year 1950 and 2010, the time series plot indicated high departures of income from the LS.

Thus, in the two respective years it characterises positive autocorrelation. In the year 1970 and 1990, the time series plot experienced a negative autocorrelation characterized with low tendencies in the departure values. In conclusion of the autocorrelation of LS, the time series plot indicates the change in autocorrelation from positive to negative with a sequence of 30 years. For example from 1950 to 1970, the time series plot showed high tendencies that translate to persistence. Between 1970 and 1990, the time series plot shows little trends showing small tendencies that bring to negative autocorrelation.

From 1990 to 2010, the time series show a high tendency for positive autocorrelation. Thus, it follows the time series that the change in degree of bias happens for a period of 20 years. Hence, provides chances of possible prediction of the LS with a 95% confidence band. Test the series Lrc, Lrpdi and Ls (=Lrc - Lrpdi) for being (1) versus (0). What happens if you test Ls on the periods 1947-1980 and 1980-2009 separately?

Plot the time series for Ls and interpret your results. The analysis of autocorrelation of RPDI, autocorrelation of DL RPDI, and autocorrelation of LRPDI with reference to 95 per cent confidence band indicates the change in computation intervals of the lagged autocorrelation. For example, the autocorrelations of RPDI, LRPDI, and DL RPDI have a decreasing lagged autocorrelation from the year 1950 to 2010 having the lowest value. The variance of the log -1 autocorrelation in the year 1950 depends on the autocorrelation coefficients of year 1970 to 2010.

The autocorrelation indicated by significantly lagged coefficients means that the theoretical autocorrelation appears to fall towards the value of zero. However, in the case presented in the data, the autocorrelation seem to be reducing in the standard error of the significant lag but not towards zero value. Thus, the autocorrelation for the data of RPDI, LRPDI and DLRPDI has random series. In reference to Bartletts formula that deals with autocorrelations of nonlinear processes it integrates the use of asymptotic data distribution. Thus, the data presented in the three autocorrelations has a decreasing standard error in the coefficients due to because it comprises of asymptotic data distribution. Estimate alternative Univariate models for the differences (DLrpdi, Dlrc) using observations up to 1999q3, leaving the remaining observations for out-of sample forecasting.

Start with AR models, and then you can try different mixed ARMA specifications. AR model: Arma model (q) Regression includes one of the Univariate model provided they have one variable that will give mean of the single variable as fitted value. The autocorrelation used tools such as time series plot and lagged scattered plot to determine the autocorrelation of data presented.

Time series plot has two forms of indications namely active and negative forms. References WEI, W. W. S. (2005). Time series analysis: univariate and multivariate methods. Boston, MA [etc. ], Pearson Addison Wesley. Brockwell, P. J., & Davis, R. A. (2002). Introduction to time series and forecasting. New York: Springer. WRAGG, T. (2011). Univariate and Multivariate Methods for the Analysis of Repeated Measures Data. München, GRIN Verlag GmbH. MONAHAN, J. F. (2001). Numerical methods of statistics. Cambridge, Cambridge University Press. GUERARD, J. (2012). Introduction to financial forecasting in investment analysis. New York, Springer.

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