Time series Regression : Descriptive Methods Jun Choi Ghiyoung Im AGENDA Overview › Definition › Examples › Problems & Treatment Pros & Cons of Time-Series Regression Application to IS Field References TIME SERIES • • A collection of data Xt (t = 1, 2, …, T) with the interval between Xt and Xt+1 being fixed and constant. Interested in not only the particular values of the observations but in the order in which they appear. • The purpose 1) Find the particular mechanism to use in forecasting the future. 2) Put that mechanism to use in forecasting the future. TIME SERIES - A time series is a sequence of observations which are ordered in time (or space). If observations are made on some phenomenon throughout time, it is most sensible to display the data in the order in which they arose, particularly since successive observations will probably be dependent. Time series are best displayed in a scatter plot. The series value X is plotted on the vertical axis and time t on the horizontal axis. - Time series regression models are specially suitable for evaluating short-term effects of time-varying exposures. In time-series studies, a single population is assessed with reference to its change over the time. EX 1 A firm’s performance (Y) is related to its IS investment (X1) and Marketing intensity (X2), for the past 16 years. Nonlagged Model. The regression model for Example 1 involves coincident time series ;that is the two indicator series refer to the same time period as the dependent variable series, Assuming that the effect of the independent variables are linear and additive, the time series regression model is as follows. Yt 0 1 X t 1 2 X t 2 t . Here, Yt is a firm’s performance, Xt1 is the IS investment, and Xt2 is the consumer price index. each for year t EX 2 The size of organization (y) is related to the amount of IS investment five years earlier (x) for the past 20 years. Lagged Model. Sometimes , it is possible to construct a regression model in which some or all of independent variables are lagged. Example 2 is an illustration, the following simple linear regression model might be useful. Yt 0 1 X t 5 t Here, Yt is the size of organization in year t and Xt-5is the amount of IS investment five years earlier. This equation represents a lagged time series, the lag that begins five years. PROBLEMs that may occur in Time Series MULTICOLLINEARITY – One independent variable is excessively linearly correlated with another independent variable HETEROSCEDASTICY – The error terms don’t have a constant variance AUTOCORRELATION – Error terms are correlated through time AUTOCORRELATED ERROR TERMS Autocorrelated error terms is when the error terms are correlated with each other. This is only a consideration when the model is a "time series" model. There are three treatments for autocorrelated error terms: 1. finding an important omitted variable 2. transforming the variables based upon generalized least squares 3. introducing Time as a variable on the right hand side of the equation THE DURBIN-WATSON STATISTIC The Durbin-Watson Statistic is used to test for the presence of first-order autocorrelation in the residual of a regression equation. The test compares the residual for time period t with the residual from time period t-1 and develops a statistic that measure the significance of the correlation between these successive comparisons. The formula for the statistic is : d (et et 1) t 2 n et t 1 n Where : d = Durbin-Watson statistic e = residual (Yi – Ye ) t = time period counter The statistic is used to test for the presence of both positive and negative correlation in the residuals. The statistic has a range of from 0 to 4, with a midpoint of 2. The Null Hypothesis ( H0 ) is that there is no significant correlation. PROS & CONS of TIME-SERIES STRENGTHS – Validate the relationship over time. (Consistency, Specification, Transformation) – Explain the past and forecast the future. WEAKNESSES – Time, money, and energy matter – Still ambiguous in context and process APPLICATION to IS Field • Motivation: Productivity paradox • Firm-level investment in IT and corresponding productivity • Underlying theory: economic theory (Cobb-Douglas & CES production function, Tobin’s q, etc.) APPLICATION Example * • CD function: Q= (IL, L, IK, K) Q: Output, IL: IT labor, IK: IT capital • CD function: ln(Q)ij = 1ln(IL)ij + 2ln(L)ij + 3ln(IK)ij+ 4ln(K)ij : elasticity of each of the input factors i: individual firm, j: year * Source: Kudyba ISR2002 APPLICATION Example * IT Capital Regression Results (CD Prod Function) Output (Sales) 1995 Parameters Coeff. (t-stat) IT Labor .228 (4.92) Non-IT Labor .306 (11.3) Capital .227 (9.97) IT Capital .122 (2.93)*** N Observations 348 *** Significant at the .01 level 1996 Coeff. (t-stat) .223 (4.43) .244 (8.40) .244 (10.44) .163 (3.32)*** 355 1997 Coeff. (t-stat) .094 (1.63) .392 (10.41) .232 (7.59) .184 (3.24)*** 188 REFERENCES • Ostrom Jr. (1990). Time series regression. Beverly Hills, CA: Sage. • Nelson (1973). Applied Time Series Analysis. San Francisco: Holden-Day. • McCleary & Hay, Jr. (1980). Applied Time Series Analysis for the Social Sciences. Beverly Hills, CA: Sage. • Kudyba (2002). Increasing returns to information technology. ISR. • Hitt & Brynjolfsson (1996). Productivity, business profitability, and consumer surplus: Three different measures of information technology value. MIS Quarterly. • Bharadwaj et al. (1999). Information technology effects on firm performance as measured by Tobin's q. Management Science.
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