Forecasting using exponential smoothing: Developing understanding via replication and contemporary applications

1. Abstract

This case study presents and illustrates single exponential smoothing and Holt’s linear method via their application to topical economic/financial series. It is argued that application of the methods is vital to develop a full understanding of their structure and nature. Further, it is argued that a deeper understanding of the methods is achieved as a result of replicating results generated automatically via software packages, with the mastery of techniques developed as a result of practical experience of the equations and issues associated with the methods.

2. Introduction

Exponential smoothing is a familiar feature of forecasting modules in business, economic and finance. While the broad heading of exponential smoothing covers a range of methods including some recent developments, the long established approaches of single exponential smoothing and Holt’s linear methods still occupy central positions within this area of analysis. In this case study it is argued that given the algebraic nature of these approaches, their demonstration via application to relevant series is vital to engage students and allow the development of a mastery of these methods. As a consequence, topical applications are presented herein to allow the nature, structure and relevance of the methods to be demonstrated.

To illustrate the various important issues associated with the smoothing methods considered (the structure of the relevant component equations; the selection of smoothing parameter values; initialisation of the methods; the generation of forecasts), the topical example considered concerns house prices and house price inflation within a London borough (Lambeth). As result of considering a trending house price series and its associated non-trending inflation rate, the analysis emphasises the suitability of alternative methods for time series with different properties.

This case study proceeds by discussing single exponential smoothing and Holt’s linear method in turn with an accompanying empirical application to the house market provided for each. In addition to the automatic creation of results from application of the smoothing methods using EViews, the analysis involves replication of the results using Excel. It is argued that this replication provides a step-by-step consideration of the methods which allows a deeper understanding of the approaches, their underlying nature and their means of application.

3. Holt’s linear method

In contrast to the inability of single exponential smoothing to capture trending behaviour, Holt’s linear method incorporates an explicit trend, or slope, term within its structure to permit the analysis of series with either upward or downward trends. The three equation approach employed under Holt’s linear method involves equations for the level (L_t) and slope (b_t) along with a forecasting equation. While the level provides a smoothed estimate of the location of a series, the slope provides a smoothed estimate of the change, or trend, of a series. The specific equations are given as below:

1) L_t = αY_t + (1 – α)(L_{t – 1} + b_{t – 1})

2) b_t = β(L_t – L_{t – 1}) + (1 – β)b_{t – 1}

3) F_{t + m} = L_t + b_tm

where Y_t denotes the variable of interest and m denotes the forecast horizon. Initial issues that arise when considering the above equations are the initialisation of the method when both the level and the slope terms require values from earlier periods and the determination of the values of the smoothing parameters (α, β). With regard to the issue of initialisation, various approaches are available. A simple approach for the level is to set its first value equal to the first value of the series (L₁ = Y₁). Considering the initialisation of the slope, various approaches are employed in practice, with EViews employing a measure of the average change across the first half of the sample considered (further details on this issue are provided in the accompanying Excel file discussed below).

To illustrate the nature of Holt’s linear method via application, data available from the EViews file Lambeth.wf1 are utilised. This file contains monthly, seasonally adjusted observations on house prices and house price inflation for the London borough of Lambeth over the period 1995 to 2015. Two series are provided, these being the natural logarithmic values of house prices (h) and their first difference (dh), with the latter providing a measure of house price inflation. As to be discussed later, the nature of single exponential smoothing and its weighted averaging of sample observations means that is it is not appropriate for series exhibiting trending behaviour, but rather series displaying fluctuation about an underlying constancy. As such, single exponential smoothing is applied to the house price inflation series dh. However, as trending behaviour is addressed explicitly under Holt’s linear method, it can be applied to the trending house price series h.

Using EViews, Lambeth house prices are examined with the full range of data available partitioned to allow the creation of both one-step and multi-step ahead forecasts. More precisely, January 1995 to December 2013 is employed as a sample period, with January 2014 to December 2015 as a forecast period. While one-step ahead within-sample forecasts are created over the period 1995-2013, the forecast period 2014-2015 allows the generation of multi-step ahead forecasts from a 2-step ahead forecast for February 2014 through to a 24-step ahead forecast for December 2015. These 1-step and multi-step forecasts are denoted as hsm in the EViews file. Summary results from EViews are available in Table One below where, amongst other items, the values of the smoothing parameters (α, β), determined on the basis of minimising the mean square error of the within-sample forecasts, are provided.

Table One: Holt’s Linear Method

Sample: 1995M01 2013M12 Included observations: 228 Method: Holt-Winters No Seasonal Original Series: h Forecast Series: hsm
Parameters:	Alpha		0.8200
	Beta		0.6600
Sum of Squared Residuals			0.017004
Root Mean Squared Error			0.008636
End of Period Levels:		Mean	12.97841
		Trend	0.014225

While an ability to appreciate and interpret the findings in Table One is of importance, replication of the derived values of hsm is particularly useful to gauge understanding of the method. As a result, the Excel file exponential.xlsx is provided to demonstrate such replication, with the first spreadsheet in this file replicating the hsm series via use of the smoothing parameters in Table One and equations (1) to (3). In addition to providing the relevant results, further information is provided within the spreadsheet to explain the replication undertaken. As the file shows, while 1-step ahead forecasts are produced in general with m = 1 in (3), for the period February 2014 to December 2015, multi-step forecasts are produced with m = 2,3,...24 as appropriate.

4. Single exponential smoothing

A standard presentation of single exponential smoothing takes the form of an adjustment process where the forecast for the next period (F_{t + 1}) is specified as the current forecast (F_t) plus an ‘element’ of the current forecast error (Y_t – F_t). This is expressed in (4) below.

4) F_{t + 1} = F_t + α(Y_t – F_t)

Clearly, the degree of adjustment, or the ‘element’ of the current forecast error used, is dependent upon the value of the smoothing parameter (α) which is bounded by (0,1). Minimal re-arrangement of (4) results in the expression (5) below:

5) F_{t + 1} = αY_t + (1 – α)F_t

Via the application of repeated substitution to (5), F_{t + 1} can be re-expressed as:

6) F_{t + 1} = αY_t + (1 – α)αY_{t – 1} + (1 – α)²αY_{t – 2} + ... + (1 – α)^{t – 1}αY₁ + (1 – α)^tF₁

From inspection of (6) it can be seen that all observations of Y_t are employed in the generation of the forecast F_{t + 1}, but the weights employed are subject to exponential decay as a result of the bounded nature of α.

To illustrate the nature of single exponential smoothing via application, the Lambeth house price inflation series (dh) available from the EViews file Lambeth.wf1 is utilised. Summary results from EViews are available in Table Two below where smoothing parameter (α), determined on the basis of minimising the mean square error of the within sample forecasts, is provided. The resulting smoothed series, or one-step ahead forecasts, is denoted as dhsm in the EViews file.

Table Two: Single Exponential Smoothing

Sample: 1995M02 2015M12 Included observations: 251 Method: Single Exponential Original Series: dh Forecast Series: dhsm
Parameters:	Alpha				0.4560
Sum of Squared Residuals					0.019887
Root Mean Squared Error					0.008901
End of Period Levels:				Mean	0.005462

To ensure a deeper understanding of the nature of single exponential smoothing, the generated forecast series dhsm can be replicated using the original inflation series dh and the optimised α. To generate the forecasts F_t + 1, a value for F_t is required, as shown by (4) and (5). As a consequence, there is an issue of initialisation as a result of the need to specify the first forecast (F₁). While alternative initialisation options exist such as setting the first forecast equal to the first actual value (F₁ = Y₁), EViews employs an averaging procedure across approximately the first half of the sample observations for Y_t. With the optimised α and the initialisation method known, the original series can be employed to create the forecast series. As with Holt’s linear method previously, the required replication of dhsm is provided in the Excel file exponential.xlsx with both results and explanatory information provided. Again, the calculations can be viewed to explain the calculation of the forecast series and hence the manner in which single exponential smoothing is applied.

5. Conclusion

The current case study has argued that a more complete understanding of exponential smoothing can be achieved via a more practical delivery involving application and replication. To that end, the necessary material to support this has been provided via consideration of a topical example involving the London housing market.

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