Teaching Note 6

Analyzing the Business Environment

Time-Series Analysis

An Example: Forecasting GNP

To illustrate univariate economic forecasting, the most frequently used indicator of the economy, Gross National Product (GNP}, is used as an example.. GNP measures total value of goods and services produced annually. In Table 1 (below), are GNP data for the years 1960 - 1983, along with calculations needed to assess future values of GNP. Looking at the data, if a guess were made of GNP after 1983, an observer could reason that (1) GNP seems to be increasing and that (2) future GNP should be higher than that in 1983. There is an uneasiness about this guess, however, because the historical data also show that there are years for which GNP declines. These troughs are economic recessions. As illustrated in the graph, Figure 1, after GNP declines there is an upturn in GNP, indicating economic recovery. The ups and downs of GNP indicate cyclical variations in the economy that impact on business. Trying to forecast a cyclical economy with the available, simple linear regression model can result in problemsome predictions as straight line models do not accommodate the effects of business cycles.

Linear regression of the data does provide information about direction of GNP. A positive slope for GNP evidences a growing economy. A negative slope evidences a declining economy. More typically, we are interested in how the economy is doing for discrete periods of time. With our data one can be confident that the overall economy is not in decline, but still not rule out a pending cyclical recession. We would like to be able to extrapolate from the historical observations how past cyclical variations affect future GNP movements. This requires that we move from the simple straight line regression model.

Time-series analysis introduces techniques that permit us to investigate the cyclical movements of GNP. The data are analyzed to identify three type of components:

• a trend component - this is the direction of the observations across time
• a cyclical component - this is the variations of the observations that give rise to troughs and peaks of a seasonal component - systematic variations of the data explained by regular, seasonal processes (not illustrated in this handout)
• an irregular component - this is the unexplained variations in observations

In the illustration of forecasting GNP, the time-series model is:

GNP = (trend value) + (cyclical component) + (irregular component)

Trend Component.- The trend component can be identified by simple linear regression. Figure 1 graphic illustrates the linear model to identify direction of the observed GNP.

Table 2. Five Year Moving Average: Cyclical Trend Analysis

The trend component can be identified also by examining the moving average. If we were interested in determining if GNP in 1962 were above or below the trend, 1962 GNP could be compared with a 5 year average of 1960, 1961, 1962, 1963, and 1964. The AVERAGE = (737.2+756.6+800.3+832.5+876.4) ÷ 5 = 800.6. The ratio of 1962 to this average is 99.96, slightly below the 5 year trend. In Table 2 the 5 year moving average is calculated for 1962 - 1981. Note that the moving average cannot be calculated for 1960 and 1961 - longer periods lose more data points and shorter periods tend to be affected by the cyclical and irregular components. In Figure 2 the graph shows actual values of GNP compared with its moving trend values

Cyclical Component.- The cyclical component of time series data is patterns produced when subsequent observations are related to previous observations. This component can be illustrated by two simple analytic techniques:

(1) Examine in Table 1 the ratio of the linear model data (Col. 3) to the actual data (Col. 2). This ratio is calculated in Col. 4 and plotted in Figure 3, below.
(2) The trend component also can be demonstrated by examining regression residuals. Residuals are the difference between what a linear model predicts for GNP each year and actual GNP. Residuals are plotted in Figure 4, below.

The graphs are not identical because of the methods; but, both methods reveal underlying cyclical patterns in the GNP data.

Irregular Component.- In time series data, as is the case with all statistical analysis of data, there is error in fitting the data exactly. In time series the error is termed the irregular component. These are deviations in predictions of the data that cannot be explained by an analysis of trend and cyclical effects of the series.Figure 3. Plot of Ratio of Linear Estimate to Actual GNP:Cyclical Component

Econmetric Modeling

An Example: Klein's Model I of U.S. Economy

Econometric modeling is an advanced application of multivariate regression in which there are more than one independent variable. In this course we can only illustrate its application, as advanced techniques are required for solution. A model of an economy is constructed using several equations each representing a characteristic or factor that is assumed to influence economic behavior. Such a model is termed a simultaneous equation analysis because several equations are evaluated at the same time in the model. The model is more of a system of interdependent equations that are solved. Econometric models are usually characterized by:

• the use of time-series data
• dynamic states, in which changes over time play an important role, are moideled by sets of equations, called identities, by which important factors are specified or defined by equations

Klein's Model I of the U.S. economy is a typical textbook illustration of econometric models. It models the U.S. economy in the period between the two World Wars as a system of 8 equations (all variables except Atime _t are measured in billions of 1934 dollars):

Outcome (y) variables that are statistical relationships:

y₁ : Aggregate Consumption, C_t. = a₁P_t + a ₂P_t-1 + a ₃ W_t + e₁
y₂ : Net Investment, I_t = b₁P_t + b₂P_r-1 + b₃K_t-1 + e₂

y₃ : Private Wage Bill, W_t^* = c₁E₁ + c₂ E_t-1 + c₃A_t + e₃

Outcome (y) variables that are identities or defined by mathematic relationships:

y₄ : Total Profits, P_t = Y_t - W_t
y₅: Total Income, Y_t = C_t + I_t + G_t - T_t
y₆ : End-of-Year Capital Stock, K_t = K_t-1 + I_t
y₇ : Total Wage Bill, W_t = W_t^* + W_t^**
y₈ : Total Production of Private Industry, E_t = Y_t + T_t - W_t^**

x₁: Government Wage Bill, W_t^**
x₂: Taxes, T_t
x₃: Government Non-Wage Expenditures, G_t
x₄: Time in Years from 1931, A_t

The system of equations is used to estimate the 9 parameters: a₁ , a₂, a₃ , b₁ , b₂, b₃ , c₁ , c₂, c₃