Load The Data In Daily IBM Data Using IBM Scan Daily

Load The Data In Dailyibmdat Using The Command Ibm Scandailyi

Load the data in “dailyibm.dat” using the command ibm. This series is the daily closing price of IBM stock from January 1, 1980, to October 8, 1992. The analysis involves various steps to assess the stationarity of the time series and prepare it for modeling.

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The process of analyzing the IBM stock closing price data over the specified period involves a careful examination of the series’ behavior, particularly concerning stationarity—a key property for many time series models. The initial step is to load the data and visualize it. Plotting the raw data provides a visual impression of its trend, variability, and potential non-stationarity. An accompanying autocorrelation function (ACF) plot helps identify serial dependencies and the presence of any autocorrelation at various lags, bolstering the interpretation of stationarity.

In the first analysis, after plotting the raw data, the series appears to exhibit notable trends and heteroscedasticity, indicating non-stationarity. The ACF plot often shows slow decay, further confirming the series' dependence structure and trend. These visual cues suggest the need for transformation or differencing to achieve stationarity, which is critical for accurate modeling and forecasting.

To address non-stationarity, the first common approach is differencing the data. The command diff(ibm) computes the first difference, which removes linear trends and stabilizes the mean. Plotting this differenced series often reveals a more stationary pattern, with less obvious trends and a more rapid decay in the ACF plot. Nonetheless, confirmatory tests, such as the Augmented Dickey-Fuller test, can substantiate these visual assessments. Differencing tends to be effective in stabilizing the mean, especially when a linear trend is present.

Another strategy to achieve stationarity, especially if the data exhibits exponential trends or heteroscedasticity, is to apply a logarithmic transformation. Using log(ibm), we transform the data, which can stabilize variance and, in some cases, remove multiplicative trends. Plotting the logged data typically reveals a less pronounced trend and more stabilized variance. Consequently, the series may appear more stationary, although differencing the logged data often yields a better stationary series.

Combining transformations—such as taking the difference of the logged data—can sometimes produce even more stationary data. Directly taking log(diff(ibm)) is generally discouraged because the differenced data may contain zero or negative values, making the logarithm invalid or undefined. Instead, the recommended approach is to difference the log-transformed data, i.e., difflogibm. This transformation often results in a series that appears stationary, with an ACF that decays rapidly and no visible trends or heteroscedasticity, aside from a few outliers.

Outliers can distort the analysis, so filtering extreme values is helpful. The command difflogibm -0.1] removes extreme negative outliers. Plotting this cleaned series usually shows more stable behavior, and the ACF plot confirms the absence of significant autocorrelations, indicating approximate stationarity.

In long series like this, certain segments may exhibit different behavior or structural shifts. Dividing the series into two parts, for example, difflogibm1 and difflogibm2, allows separate analyses. Visual comparisons and ACF plots highlight differences in stationarity or dependence patterns across the segments. Usually, earlier parts of the series may show more pronounced trends or volatility, whereas later parts may stabilize.

Finally, assuming that the model for the second part, difflogibm2, follows a simple linear form with Gaussian white noise, i.e., dt = δ + wt, is reasonable if the series appears stationary and residuals are uncorrelated. Estimating the parameters δ and σw involves calculating the mean and standard deviation of the residuals from the fitted model. The estimates provide insights into the average change per period and the variance of noise, crucial for subsequent time series modeling.

References

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