Time Series Analysis and Adjustment

Measuring, Modelling and Forecasting for Business and Economics

Haim Y. Bleikh, Warren L.Young

• Publisher: Routledge
• Year: 2016
• Topic: Business
• eBook ISBN:
  9781317010173
• Subtopic: Business General
• Index: Business

In Time Series Analysis and Adjustment the authors explain how the last four decades have brought dramatic changes in the way researchers analyze economic and financial data on behalf of economic and financial institutions and provide statistics to whomsoever requires them. Such analysis has long involved what is known as econometrics, but time series analysis is a different approach driven more by data than economic theory and focused on modelling. An understanding of time series and the application and understanding of related time series adjustment procedures is essential in areas such as risk management, business cycle analysis, and forecasting. Dealing with economic data involves grappling with things like varying numbers of working and trading days in different months and movable national holidays. Special attention has to be given to such things. However, the main problem in time series analysis is randomness. In real-life, data patterns are usually unclear, and the challenge is to uncover hidden patterns in the data and then to generate accurate forecasts. The case studies in this book demonstrate that time series adjustment methods can be efficaciously applied and utilized, for both analysis and forecasting, but they must be used in the context of reasoned statistical and economic judgment. The authors believe this is the first published study to really deal with this issue of context.

CHAPTER 1
LITERATURE SURVEY

The investigations of time series and seasonal adjustment methods have their origins in the nineteenth century. The idea that a time series is made up of both observed and unobserved components was known in scientific fields, such as astronomy and meteorology. In the middle of the nineteenth century, the Dutch meteorologist, Ballot, averaged temperature data for different periods of time (monthly, weekly etc.) in order to isolate individual periodic components that he assumed to be dependent upon astronomical factors, such as the moon’s location (Grether and Nerlove, 1970).

At the beginning of the twentieth century, there was much research concerning seasonal adjustment. One of the most important works in the field was done by Persons in 1919 (Bell and Hillmer, 1984). He was probably among the first who came up with the idea of decomposition of time series into four components – trend, cyclical, seasonal and irregular components. Persons’ method is known as the link-relative method, which isolated the components by utilizing de-trended and seasonally adjusted data to construct business indices. Despite the fact that Persons believed that seasonal variation is not fixed over time, in his methodological framework he utilized fixed seasonal factors for adjustment of the data. However, the use of fixed seasonal factors, as was suggested by Persons was argued by different researchers. It appears that the first researchers who introduced varying seasonal factors into seasonal adjustment procedure were Sydenstricker and Britten in 1922 (Fischer, 1995). However, it must be noted that the recognition of changing seasonality and its applications, had already been introduced in the nineteenth century by Gilbart in 1854, and at the beginning of the twentieth century, by Jevons in 1909 (see for example Grether and Nerlove, 1970; Bell and Hillmer, 1984). Persons’ idea motivated others to continue research on the seasonal adjustment problem during the 1920s and in particular, these methods were proposed by other writers, such as Hall, Gressens, Clendenin, and Joy, and Thomas (Bell and Hillmer, 1984). In general, it was agreed that:

a) seasonality changes over time;

b) there is a need to account for trends and cycles when estimating the seasonal component;

c) trends and cycles cannot be described by mathematical formulae alone;

d) extreme observations had to be dealt with.

In 1931, Macauley suggested a procedure known as the ratio to moving average method, for seasonally adjusting data. This method was very important, since it became the basis for the Census Method 1 program, first introduced in 1954. Macauley’s method was based on moving averages, rather than fitting explicit functions to the data, a technique that was borrowed from actuaries (see for example Shiskin et al., 1967, Hillmer et al., 1983).

On a parallel track, in the 1920s, the Russian mathematician and economist Slutzky, showed that utilizing a white noise sequence with a suitable weighting of random numbers, can create almost any pattern desired. There is, in fact, a theorem due to Slutzky (1937), which shows that, under certain conditions, repeated application of moving averages can eventually lead to a sequence that follows a sine wave. This is a highly important result, since it is well known that moving averages are applied in order to smooth the data, which indeed, can be said to be the exact opposite of Slutzky’s results (Gilchrist, 1976). The models generated from this process are called moving average models, since they are created by averaging white noise sequence. At about the same time of Slutzky’s work, Yule (1927) discovered the same fact, but by the reverse process. In particular, he showed that time series data could be described by a linear aggregate of previous values of the process and a current error term. These models are called autoregressive models (further discussion from the historical point of view of contributions of Slutzky and Yule can be found in Gottman, 1981).

During the 1950s and continuing over the next decades, in particular after the fundamental book of Burns and Mitchell (1946) at NBER, much attention had been addressed to analyzing business cycles and related macroeconomic factors. In their research, Burns and Mitchell concatenated a large number of statistical indicators of recessions and expansions into one signal of turning points in the US business cycle. However, their work led to methodological conflict, and in 1947 their work was highly criticized by Koopmans, a leading figure on the Cowles Commission and later Nobel Laureate. He wrote a review of Burns and Mitchell’s book under the title Measurement without Theory. The review concentrated on the appropriate methodology for the analysis of business fluctuations. It opposed the NBER empirical approach, and refused to accept any theoretical basis for these statistical investigations (Malinvaud, 1983; Epstein, 1999). The stage was set, for the development of modern econometric analysis by distinguished researchers from Cowles Commission. This approach concentrated on the estimation of simultaneous equations models, hypothesis testing and estimation by the maximum likelihood technique etc. These results, for example, are collected in Hood and Koopmans (1953).

Since then, different approaches have been suggested to measure business cycles. In general, business cycles are fluctuations in output and employment about a trend. But, what is a trend and how can we measure it? These questions became important over time. A method which enjoyed great popularity among economists, assumed that a trend is a deterministic function of time. The cyclical component is simply the residual from a fitted trend line. The problem with this approach is “that the long-run evolution of the time series is deterministic and perfectly predictable” (Beveridge and Nelson, 1981: 152). Moreover, Zarnowitz and Ozyildirim (2002) asserted that too much of the overall variation is attributed to the business cycles. Thus, a linear trend does a poor job of distinguishing between good and bad economic periods over time. Another approach to measuring business cycles is known as phase average trend (PAT) method developed in NBER during the 1970s. This method is based on computing a centered 75 month moving average, which makes it necessary extrapolate backward/forward over first/last 37 months of the series covered. The use of moving average techniques in general generates an end point problem, but this is a common phenomenon with other methods as well (Zarnowitz and Ozyildirim, 2001).

Research on time series analysis was primarily concerned, on the other hand, with developing techniques for the decomposition of a time series into trend, cyclical seasonal and irregular components. Exponential smoothing proved through the years to be very useful in many analyses. These methods originated in the 1950s and 1960s with works of Holt (1957), Winters (1960) among other writers.

Until the late 1960s, there was no synthesis between econometric analysis and time series analysis, though famous works such as that of Cochrane-Orcutt (CO) (1949) and afterwards of Prais-Winsten (PW) (1954) had already been introduced in the econometric framework of time series regression. The works of CO and PW dealt with the correction of serial correlation in time series regression when there is non-zero autocorrelation between the successive errors.¹ On a parallel track, Durbin and Watson (DW) (1950, 1951), provided a test for detecting the existence of a first order serial correlation in the errors of the time series regression model. Later, Breusch (1978) and Godfrey (1978) developed a Lagrange Multiplier (LM) test that provided the econometricians with the ability to detect higher order autocorrelations.

However, this situation changed dramatically with publication of the volume by Box and Jenkins in 1970. The BJ approach is based on autoregressive-integrated moving average (ARIMA) processes. Their approach involves identifying an appropriate ARIMA process, fitting it to the data and finally using it for forecasting. One of the most important features of this approach is its ability to provide a model with an adequate description of the data.

Following the approach of Box and Jenkins, Beveridge and Nelson (1981) and Nelson and Plosser (1982) introduced new approaches related to the definition of trend in analyzing business cycles. The first approach defined trend as a long-horizon forecast, rather than one that is fixed and pre-determined. It implied that the trend shifts as new information (data) reveals more information about the future (Nelson, 2008). The second approach dealt with trend-stationary and difference-stationary processes which could be inferred from unit root analysis proposed by Dickey and Fuller (1979, 1981). In particular, it was argued by the writers that important economic aggregates have no tendency to return to a linear trend, suggesting that the trend is, rather stochastic.

As shown above, in the 1980s a sequence of influential papers dealt with measuring the macroeconomic variables in the context of business cycles. Another popular approach to the measurement of business cycles is the Hodrick and Prescott (HP) filter (1980, 1997). Indeed, Prescott wrote:

Our trend is just a well-defined statistic, where a statistic is a real valued function. Hodrick and Prescott’s trend statistic mimics well the smooth curve that economists fit through the data … A desirable feature of this definition is that with the selection of smoothing parameters for quarterly time series, there are no degrees of freedom and the business cycle statistics are not a matter of judgment (2004: 377).

As it is well known, seasonally adjusted data provides more interpretable measures of changes occurring in a given period, and reflects real economic movements without problematic seasonal changes. The choice of method for seasonal adjustment is crucial for the removal of all seasonal effects in the data. Thus, in 1954 the US Census Bureau released its first computer program for seasonal adjustment – Census Method 1. Over the years this program underwent different modifications. The last version of this program was called X-11 (Shiskin et al., 1967). In the mid-1970s Statistics Canada developed the X-11-ARIMA program, which was based on X-11 and the approach of Box and Jenkins (Dagum, 1978). The latter program, also underwent modifications, leading to the version known as X-11-ARIMA/88. During the 1990s US Census Bureau developed a new seasonal adjustment program, X-12-ARIMA, while another program in common use is the TRAMO/SEATS package, developed by the Bank of Spain (further discussion about the evolution of program methods will be discussed in Chapter 5).

The counter-argument to the utilization of seasonally adjusted data in macro-economic analysis, is derived from the fact that seasonally adjusted data do not provide consistently improved predictions, and in many cases forecasts based upon them are even less accurate than those derived from non-seasonally adjusted data. Comparative analyses related to this issue is well established in the literature and can be found, for example, in Makridakis and Hibon (1979), Plosser (1979), and Moosa and Ripple (2000). Yet, no clear cut answer is provided, as it is argued for example, by Bell and Hillmer (1984).

Throughout the years different studies dealt with the fundamental question: does it make sense to utilize filtered data? For example, Lee and Siklos (1993) tested the permanent income hypothesis (PIH) using Canadian data. Their results supported the PIH for filtered data, but not in the case where the data was not filtered. In the study conducted by Meyer and Winker (2005), a Monte Carlo simulation was performed in order to examine the performance of the HP filtered and non-filtered data in a regression framework. Their results suggested that estimates of an econometric model may be distorted by utilization of filtered data. As Nelson and Kang (1981: 742) pointed out in their paper that discussed the influence of the detrended data utilized for econometric analysis: “… the dynamics of econometric models estimated from such data may well be wholly or in part an artifact of the trend removal procedure”.

Looking back at literature, seasonal adjustment procedures were discussed in different papers. For example, Rosenblatt (1968) and Grether and Nerlove (1970) discussed the utilization of the spectral analysis when dealing with seasonally adjusting data. On the other hand, seasonal adjustment by regression analysis was discussed in Lovell (1963), Thomas and Wallis (1971) and Sims (1974).

It must be emphasized that the statistical method of time series analysis deals with univariate processes, as it is discussed in Box and Jenkins (1970). However, a substantial part of macroeconomic analysis deals with multivariate relationships. The cooptation of this statistical method into dynamic macroeconomic systems was implemented in the contributions of Granger (1969), Granger and Weiss (1983) and Engle and Granger (1987) (further discussion is given in Chapter 4).

During the 1990s, a number of important developments occurred in the area of filtering, followed the work of Hodrick and Prescott. New filters were created by Baxter and King (BK) (1995) and Christiano and Fitzgerald (CF) (1999). These filters formulate the de-trending and smoothing problem in the frequency domain. A number of empirical studies used different filters, and compared the results obtained. For example, Demertzis and Hallet (2003) and Montoya and de Haan (2007) compared CF with HP, while Guay and St-Amant (1997, 2005) provided analysis of BK and HP.

Recent developments in time series analysis and seasonal adjustment procedures are concentrated in the combination of X-12-ARIMA and SEATS. In 2006, the US Census Bureau with the support of Bank of Spain released a new program which is called X-13-ARIMA-SEATS. This program offers the user the seasonal adjustment of both methods, with improved diagnostics for the model-based seasonal adjustments (Monsell, 2006; Findley, 2005; Monsell, Aston and Koopman, 2003).

1 In such cases the OLS standard errors and test statistics are not valid. The main difference between CO and PW estimation depends on how the researcher deals with the first observation while treating the serial correlation problem.

CHAPTER 2
FORECASTING

2.1 FORECASTING: IMPORTANCE AND LIMITATIONS

The choice and application of the proper forecast technique has always been an important issue for most economic and financial institutions, and statistical bureaus. At the micro-level of the firm, an entire financial operation may rely on the accuracy of the forecasts since such information will likely be used to make interrelated decisions in areas of budgeting, purchasing, marketing and advertising, capital financing, personnel management, etc. Moreover, at the macro-level of the economy as a whole, forecast accuracy can also be crucial. For example, at the firm level, a significant over-sales forecast error may cause the firm to be burdened with excess inventory carrying costs. On the other hand, an under-sales forecast can create a loss of sales revenue. During periods when demand is fairly stable, or is unchanging or growing, or on the other hand, declining at a known constant rate, making an accurate forecast is less difficult. If, on the other hand, during its history, the firm has experienced an up-and-down sales pattern, then the forecasting task can be rather complicated. At the macroeconomic level, a shortfall in tax revenue, despite a forecast to the contrary, can lead to a budget deficit, and the consequences that follow from it.

Many everyday decisions, both professional and personal are made intuitively. This often happens when a short term response is under consideration and no time and/or resources are available for a sophisticated analysis of the data. Moreover, the decision maker may believe that his intuition, in particular, is more trustworthy than any mathematical forecasting model. The most important characteristic of this kind of intuitive model is that it is not reproducible. We can, however, understand how the intuitive decision-making process works, by interviewing the forecaster and by examining previous forecasts. In this case, it seems that we are dealing with a kind of a “black box”.

As was previously mentioned, time and resources are crucial for decision making.¹ Ideally, organizations which can afford to do so, will usually assign crucial forecast responsibilities to those departments and/or individuals that are best qualified and have the necessary resources at hand to make such forecast estimations under complicated demand patterns. Clearly, a firm with a large ongoing operation and a technical staff composed of statisticians, management scientists, computer analysts, etc., is in a much better position to select and make proper use of sophisticated forecasting techniques than a company...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• List of Figures
• List of Tables
• List of Abbreviations
• Introduction
• 1 Literature Survey
• 2 Forecasting
• 3 Univariate Time Series Analysis
• 4 Further Topics in Time Series Analysis
• 5 The Development of Seasonal Adjustment Programs
• 6 Empirical Analysis
• Conclusions
• References
• Appendix 1
• Appendix 2
• Appendix 3
• Index