The Importance of Autoregressive Integrated Moving Average Model - Coursework Example

Table of Contents Overview 2 Importance of ARIMA model 3 Aspects of the Model 4 Stationarity of the AR process 4 AR Process 4 Unit Root 4 Box-JenkinsMethodology 6 Identification 6 Diagnostic Checks 7 Criticism of ARIMA Model 7 Implication of ARIMA Model Approaches 8 Conclusion 9 List of References 10 Overview ARIMA model forecasts and evaluates equivalently spaced Univariate data of time series, intervention data and transfer function data through the Autoregressive Integrated Moving Average (ARIMA). ARIMA model predicts a value in response time series as a linear combination of past and current values of other time series as well as its own past values. ARIMA models are sometimes referred to as Box and Jenkins models as this model approach was first known as Box and Jenkins. Box and Taio (1975) were the ones who first discussed the procedure of ARIMA put to work as general transfer function model (Pankratz, 1991). ARIMA model is sometimes referred to as ARIMAX model when it includes other time series as input variables. ARIMAX was referred to as dynamic regression by Pankratz (1991). A comprehensive set of means for parameter estimation, forecasting and time series model identification is provided by ARIMA procedure as it offers much more flexibility than any other model in an ARIMAX or ARIMA model that can be analysed (Liu et al., 2001). Subset, seasonal and factored ARIMA models, are all supported by the ARIMA procedure, interrupted time series models or intervention along with rational transfer function models of any complexity and multiple regression analysis with ARMA errors. For time series modelling with features that consist of estimation and diagnostic checking, identification and forecasting steps of the Box and Jenkins method, Box and Jenkins strategy is closely followed by the design of PROC ARIMA (Gujarati, 2012). Importance of ARIMA model ARIMA and transfer function models are very effective in handling practical applications as they are considered to be the best models in the forecasting and analysis of time series data. Over several decades, vast advancements in this area of research have been accomplished in regards to both, methods and theory. Unfortunately, given the great advantage they offer, they are still not as widely used as they should be (Asteriou & Hall, 2011). Because of the time consuming nature and complexity of the ARIMA model, it is not widely used in industrial applications and other mainstream businesses. Transfer function models to achieve automatic modelling when descriptive or input variables are included, automatic ARIMA modelling capabilities can be used in combination to it (Gujarati, 2012). ARIMA models play a significant role in forecasting, as the forecasts based on ARIMA models will be regarded as baseline values for forecasting comparison in addition to its own usefulness. Forecasts obtained using ARIMA model is often compared with more complicated models such as forecasts obtained from nonlinear and multivariable time series models. The existence of outliers in the series or incorrect specification in a more complicated model are signified if the forecasts generated under complicated models are less accurate than those under an ARIMA model. No matter what methodology is eventually adopted, effective auto ARIMA modelling capability displays an important role in forecasting (Sowell, 1992). The AR operators are normally placed on the left hand side of the model in almost all time series books using ARIMA models. When a constant term is present in the model, the model expression makes it hard to supply an interpretable meaning. Therefore, it is more advantageous to place the AR operators on the right hand side of the model to make it easy to obtain an interpretable meaning (Brooks, 2008). Aspects of the Model Stationarity of the AR process In the case where previous values of the error term would contain a non-decreasing effect on the current value of the dependent variable, this will suggest that the AR model is not motionless. It would also mean that since the lag length is increasing, the coefficients on the MA process would not turn to a zero. The coefficients on the corresponding MA process decrease with lag length resulting in zero, an AR model would be stationary (Barndorff‐Nielsen, 2002). AR Process The roots of the distinguishing equation lies outside the unit circle which is greater than 1 is a test for stationarity in an AR model (with p lags), the equation would be (Brooks, 2008): Unit Root One needs to describe as testing for a “unit root” as it is needed for testing for stationarity for any variable since this is based on this same idea. AR (1) model is the most basic AR model such as the Dickey-Fuller test, on which almost all the tests for stationarity are subjected to. The characteristic equation for unit root test is as follows (Gujarati, 2012) The characteristic equation of (1-z) = 0 along the AR (1) model suggests that the root of z is equal to 1. Rather than outside it, it lies on the unit circle hence one can conclude that this is non-stationary. The potential number of roots increases with the increase in the lags of the AR model. Therefore in case of two lags, the quadratic equation producing 2 roots will be available, and they both need to lie outside the unit circle for the model to be stationary (Brooks, 2008). For an AR(1) process with a constant (μ) of the unconditional mean, is denoted by (Brooks, 2008): Excluding the constant, variance for an unconditional AR process of order 1 will be (Brooks, 2008): Box-Jenkins Methodology Based on the PACF and ACF as a means of determining the lag lengths of the ARIMA model and the stationarity of the variable in question, this is a method for estimating ARIMA models. Even though the PACF and ACF methods for laying out the lag length in an ARIMA model are commonly used, still there are many other methods depending on the information criteria that can also be used (Brooks, 2008). There are four parts of the Box and Jenkins approach. First of the model is the identification of the model, second part consists of OLS which is usually estimation, third is mostly for autocorrelation which is diagnostic checking. Fourth part of this model is forecasting (Brooks, 2008). Identification The most important part of the process is the identification of the best fitting model, where it becomes as much ‘science’ as ‘art. The very first step that can be done with the correlogram is to determine whether a variable is stationary or not. It needs to be first-differenced in case if it is not stationary. To induce stationarity, it might need to be differenced again. The next step in the ARIMA “p,l,q” model, where “l” is referred to as how many times the data is needed to be differenced to generate a stationary series, is to determine the “p” and “q” (Gujarati, 2012). The PACF, also known as Partial correlogram, is employed where the number of the non-zero points determine where the AR lags need to be included as it is needed to determine the appropriate lag structure in the AR part of the model. The ACF, also known as correlogram, is used as again the non-zero points suggest where the lags are to be included as it is needed to determine the MA lag structure. If in case the data exhibits seasonal effects, seasonal predicted variables can also be included (Brooks, 2008). Diagnostic Checks By using the Q or Ljung Box statistic, this approach is only tested for autocorrelation. One would have to go back and add more lags to the identification stage and re-specify the model if in case there is an evidence of autocorrelation. It only tells if it is too small as it fails to identify whether the model is over parameterised or too big is the only criticism of this approach (Gultekin, 1983). Criticism of ARIMA Model Some of the criticisms of ARIMA forecasting are as follows: Some of the traditionally used model identification techniques are subjective, and the reliability and validity of the chosen model rely upon the skills and experience of the forecaster. Although, this criticism applies to another modelling approach too (Kenny, Meyler, & Quinn, 1998). The economic importance of selecting ARIMA model is not clear because it does not comply with any theoretical construct or model relationships. In addition, it is not possible to run policy duplication with ARIMA model, unlike the structural model approaches (Kenny, Meyler, & Quinn, 1998). The majority forecasting is expected to be large due to the nature of ARIMA model approach. Whatever the specification used the ARIMA model approaches are solely based on fluctuations in prices and past movements and indications. Therefore, without any demand and supply variables in action it would not be expected that such models could predict a change in the market place. Thus, an ARIMA model requires demand and supply side variables to fluctuate and remain in action if forecasting has to be done. The ARIMA model approaches are “backward-looking”. They are bad at forecasting the turning points, unless a turning point reflects a return to long run equilibrium level (Kenny, Meyler, & Quinn, 1998). This concept was further asserted in the study by Stevenson (2007), which was developed in the context of the British office market. This refers to the notable weakness of the model as a matter of fact; such turning points have critical implications for the investors, fund manager and the investment consultants. However, ARIMA model approaches have proven themselves to be an effective and strong model especially when short-run inflation is forecasted. ARIMA model outperforms especially more structural models in term of short run inflation prediction. The ARIMA forecasting technique not only provides a milestone for other forecasting approaches but will also provide input into forecasting in its own benefit (Kenny, Meyler, & Quinn, 1998). Implication of ARIMA Model Approaches ARIMA model forecasts and evaluates equivalently spaced Univariate data of time series, intervention data and transfer function data through the Autoregressive Integrated Moving Average (ARIMA). Numerous models are available for forecasting economic time series. One approach, which includes the forecasting of time series only, is referred to as Univariate forecasting. Autoregressive integrated moving average (ARIMA) is a subdivision of the above model Univariate forecasting. In this model time, series is expressed in terms of past figures of itself (the autoregressive component) plus present and diminishing values of “white nose” (the moving average component). Stevenson (2007) in a paper has developed a comprehensive review of the ARIMA model. The paper has referred to various studies in order to ascertain the use-ability of the ARIMA model for the forecasting purpose using time series data. Stevenson (2007) has stated that ARIMA offers an efficient forecasting for the short-term forecasts. For example, Brooks & Tsolacos (2000), developed a study of the retail rents in the United Kingdom with the use of vector and time series model. The results suggest that the use of vector and time series models can pick up prominent factors that are useful in forecasting purposes. The variable, which are important for inclusion in the auto regression system are significantly different between the two series. This is restricted to the rent indexes. Conclusion Based on the experience and conclusion it is concluded that there are two main points to be focused. There might be many difficulties arising in model fitting and predicting patterns if the data is divided into a way differently behaving same set thus, having heterogeneity factor for sample. Usually external factors are merely responsible for such difficulties and if these cannot be acquired by the information forecasted then the result is very poor (Moser, Rumler, & Scharler, 2007). ARIMA model having many limitations can forecast in the short run condition, however; it failed to provide the result in a long run approach. Thus, ARIMA model cannot be applied as a robust model as it has certain limitations, which create hindrance in making it as a perfect model. Moreover, ARIMA requires demand and supply side variable to remain active and intact without which it cannot forecast. Therefore, model is good at indicating the predictions in the short run, however, failed to provide insight in the long run. List of References Asteriou, D., & Hall, S. G. (2011). Applied econometrics. Palgrave Macmillan. Barndorff‐Nielsen, O. E. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 64, no. 2, pp. 253-280. Brooks, C. (2008). Introductory econometrics for finance. Cambridge Books. Brooks, C., & Tsolacos, S. (2000). Forecasting models of retail rents. Environment and Planning A, vol, 32, no. 10,pp. 1825-1840. Gujarati, D. N. (2012). Basic econometrics. McGraw-Hill Education. Gultekin, N. B. (1983). Stock market returns and inflation: evidence from other countries. The Journal of Finance, vol. 38, no. 1, pp. 49-65. Kenny, G., Meyler, A., & Quinn, T. (1998). Forecasting Irish inflation using ARIMA models (No. 3/RT/98). Central Bank of Ireland. Liu, L. M., Bhattacharyya, S., Sclove, S. L., Chen, R., & Lattyak, W. J. (2001). Data mining on time series: an illustration using fast-food restaurant franchise data. Computational Statistics & Data Analysis, vol. 37, no. 4, pp. 455-476. Moser, G., Rumler, F., & Scharler, J. (2007). Forecasting austrian inflation. Economic Modelling, vol. 24, no. 3, pp. 470-480. Pankratz, A. (1991). Forecasting with Dynamic Regression Models. NY: John Wiley and Sons Sowell, F. (1992). Modeling long-run behavior with the fractional ARIMA model. Journal of Monetary Economics, vol. 29, no. 2, pp. 277-302. Stevenson, S. (2007). A comparison of the forecasting ability of ARIMA models. Journal of Property Investment & Finance, vol. 25, no. 3, pp. 223-240. Read More