Graphic Method

1. Core Concept Overview & Intuition

1.1 Introduction

A time series is a sequence of observations recorded in chronological order. Because successive observations are ordered by time, they are generally not independent, and the central task of time series analysis is to separate the systematic movement of the series from the irregular movement. The graphic method is the most elementary device for this separation: the observed values $X_t$ are plotted against time $t$ and a free-hand smooth curve is drawn through the scatter of points. The curve is intended to represent the general tendency of the series, that is, its trend.

The graphic method is attractive because it is simple, requires no arithmetic, and is flexible: the analyst can accommodate bends and changes of direction that a single rigid formula would miss. Its weakness is equally clear. The method does not measure trend in numerical form, and the position of the curve depends on the judgement of the person drawing it. Two analysts working on the same data may produce two different curves, so the method is subjective. It is therefore best regarded as an exploratory first step, to be followed by an objective procedure such as the method of curve fitting, in which trend is estimated by the method of least squares.

1.2 Learning outcomes

By the end of this unit the learner should be able to:

• define a time series and distinguish the types of time series;
• use the standard terminology of time series analysis;
• state the objectives of time series analysis;
• describe the graphic method of measuring trend and explain its advantages and limitations;
• fit polynomial, exponential and growth curves to a series by the method of least squares and by the method of three selected points;
• eliminate trend from a series and interpret the residual components;
• estimate future values of a series from a fitted trend.

1.3 Definition of a time series

> Definition (Time series). A time series is a set of observations on a variable, arranged in the order in which the observations are made, usually at equally spaced intervals of time.

The variable observed is denoted $X_t$, where the subscript $t$ is the time index. The values $X_1, X_2, \dots, X_n$ constitute the series, and $n$ is the number of observations. The essential feature that distinguishes a time series from an ordinary sample is the ordering: the observation at time $t$ is not exchangeable with the observation at time $t+1$.

1.4 Types of time series

Time series are classified according to the nature of the variable recorded and the spacing of the observations.

• Univariate and multivariate series. A univariate series records one variable over time; a multivariate series records two or more variables simultaneously, for example price and quantity of a commodity.
• Continuous and discrete series. A continuous series is recorded without interruption, such as the reading of a recording thermometer; a discrete series is recorded at separated instants, such as annual production figures.
• Series of a stock and series of a flow. A stock series measures a quantity existing at an instant, such as the number of employees on a payroll; a flow series measures a quantity accumulated over an interval, such as total output during a year.
• Series classified by interval. Observations may be recorded annually, quarterly, monthly, weekly or daily. The interval determines which components of the series can be identified.

1.5 Some terminologies

• Trend ($T_t$). The long-term general direction of movement of the series, whether upward, downward or flat.
• Seasonal variation ($S_t$). A movement that repeats itself within a fixed period of less than a year, such as a twelve-month cycle in monthly data.
• Cyclic variation ($C_t$). A wave-like movement of period longer than a year, whose turning points are not strictly periodic.
• Irregular or random component ($I_t$ or $e_t$). The residual movement left after trend, seasonal and cyclic effects have been accounted for.
• Additive and multiplicative models. The components are combined either as $X_t = T_t + S_t + C_t + I_t$ or as $X_t = T_t \times S_t \times C_t \times I_t$.
• Trend values. The values $\hat{X}_t$ generated by a fitted trend curve at the observed time points.
• Elimination of trend (detrending). The removal of the trend component from the series, leaving the other components.
• Lag. The distance, in time units, between two observations being compared.

1.6 Objectives of time series analysis

• Description. To summarise the past behaviour of the series in a compact form.
• Explanation. To identify and isolate the components of the series, so that the contribution of each can be understood.
• Prediction and forecasting. To extrapolate the fitted trend, or the fitted model, to future time points.
• Control. To compare current performance with an expected path and detect departures from it.
• Comparison. To compare the behaviour of two or more series on a common scale.

1.7 The graphic method in context

The graphic method belongs to the family of procedures for measuring trend. It is preceded by the simple inspection of the plotted series and is followed by objective curve fitting. Its role is to suggest the form of the trend — linear, quadratic, exponential or S-shaped — which is then estimated by least squares or by the method of three selected points. The intuition is that a smooth curve drawn through the scatter averages out the short-term fluctuations and leaves the long-term movement visible.

2. Mathematical Formalization & Core Definitions

2.1 The graphic method

> Definition (Graphic method). The graphic method of measuring trend consists of plotting the observed values $X_t$ against time $t$ on a rectangular coordinate system and drawing a free-hand smooth curve through the plotted points, the curve being taken as representing the trend of the series.

The horizontal axis carries time $t$ and the vertical axis carries the variable $X_t$. The curve is drawn so that it passes through the middle of the scatter, with the vertical deviations of the points above and below the curve roughly balancing. The method is simple and flexible, but it does not measure trend numerically and it is subjective.

2.2 Method of curve fitting: polynomials

The objective method fits a mathematical curve $X_t = f(t)$ by the method of least squares, choosing the parameters so that the sum of squared deviations

$$
\sum_{t=1}^{n}\left(X_t - f(t)\right)^2
$$

is minimised. Differentiating with respect to each parameter and equating to zero produces the normal equations, which are solved simultaneously.

Straight line trend. For the line

$$
X_t = a + bt
$$

the normal equations are

$$
\sum_{t=1}^{n} X_t = na + b\sum_{t=1}^{n} t
$$

$$
\sum_{t=1}^{n} tX_t = a\sum_{t=1}^{n} t + b\sum_{t=1}^{n} t^2
$$

These two equations are solved simultaneously for $a$ and $b$. The constant $a$ is the intercept and $b$ is the slope, that is, the amount by which the trend changes per unit of time.

Second degree polynomial (quadratic trend). For the curve

$$
X_t = a + bt + ct^2
$$

the normal equations are

$$
\sum X_t = na + b\sum t + c\sum t^2
$$

$$
\sum tX_t = a\sum t + b\sum t^2 + c\sum t^3
$$

$$
\sum t^2X_t = a\sum t^2 + b\sum t^3 + c\sum t^4
$$

These three equations are solved simultaneously for $a$, $b$ and $c$.

Coding of the time variable. When the time points are equally spaced, the arithmetic is greatly reduced by shifting the origin to the middle of the series and rescaling. The coded time variable is

$$
t' = \frac{t - t_0}{h}
$$

where $t_0$ is the chosen origin (usually the middle year or the mean of the time points) and $h$ is the spacing between successive time points. With this coding the coded values are symmetric about zero, so that $\sum t' = 0$ and, for an odd number of observations, $\sum t'^3 = 0$. The normal equations then simplify dramatically. For the straight line the equations reduce to

$$
\sum X_t = na'
$$

$$
\sum t'X_t = b'\sum t'^2
$$

so that $a' = \bar{X}$ and $b' = \dfrac{\sum t'X_t}{\sum t'^2}$. The fitted values are unaffected by the coding, and the original parameters are recovered from $a'$ and $b'$ by reversing the transformation.

2.3 Exponential curve

For the exponential curve

$$
X_t = ab^t
$$

taking logarithms gives

$$
\log X_t = \log a + t\log b
$$

which is linear in the logarithms. Writing $Y_t = \log X_t$, $A = \log a$ and $B = \log b$, the model becomes

$$
Y_t = A + Bt
$$

The method of least squares is applied to the transformed data $(t, Y_t)$ to estimate $A$ and $B$, and the original parameters are recovered as $a = 10^{A}$ and $b = 10^{B}$ when common logarithms are used, or as $a = e^{A}$ and $b = e^{B}$ when natural logarithms are used. The exponential curve is appropriate when the series changes by a roughly constant percentage rather than a constant absolute amount.

2.4 Growth curves and the method of three selected points

Growth curves contain more parameters than can be handled by the normal equations of least squares in the usual way, since the number of parameters exceeds the number of variables available for the standard formulation. They are therefore fitted by the method of three selected points. Three ordinates $X_1$, $X_2$ and $X_3$ are chosen at three equidistant time points $t_1$, $t_2$ and $t_3$, so that

$$
t_2 - t_1 = t_3 - t_2
$$

The three ordinates supply three equations, which are sufficient to determine the three parameters of the curve.

Modified exponential. The curve is

$$
X_t = a + bc^t
$$

Substituting the three selected points gives

$$
X_1 = a + bc^{t_1}, \qquad X_2 = a + bc^{t_2}, \qquad X_3 = a + bc^{t_3}
$$

Subtracting successive equations gives

$$
X_2 - X_1 = bc^{t_1}\left(c^{t_2 - t_1} - 1\right)
$$

$$
X_3 - X_2 = bc^{t_2}\left(c^{t_3 - t_2} - 1\right)
$$

Dividing the second by the first, and using $t_2 - t_1 = t_3 - t_2$, gives

$$
\frac{X_3 - X_2}{X_2 - X_1} = c^{t_2 - t_1}
$$

from which $c$ is obtained. Substituting $c$ back into the difference equations gives $b$, and substituting $b$ and $c$ into any one of the original equations gives $a$.

Gompertz curve. The curve is

$$
X_t = ab^{c^t}
$$

Taking natural logarithms on both sides gives

$$
\ln X_t = \ln a + c^t\ln b
$$

Writing $Y_t = \ln X_t$, $A = \ln a$ and $B = \ln b$, this becomes

$$
Y_t = A + Bc^t
$$

which is a modified exponential in the logarithms. The method of three selected points is therefore applied to the transformed ordinates $Y_1$, $Y_2$ and $Y_3$ exactly as for the modified exponential, and the parameters $a$ and $b$ are recovered by exponentiation.

Logistic curve. The curve is

$$
X_t = \frac{k}{1 + e^{a + bt}}
$$

where $a$, $b$ and $k$ are constants. Taking three ordinates at equidistant time points gives

$$
X_1 = \frac{k}{1 + e^{a + bt_1}}, \qquad X_2 = \frac{k}{1 + e^{a + bt_2}}, \qquad X_3 = \frac{k}{1 + e^{a + bt_3}}
$$

Rearranging each equation gives

$$
a + bt_1 = \log\left(\frac{k - X_1}{X_1}\right)
$$

$$
a + bt_2 = \log\left(\frac{k - X_2}{X_2}\right)
$$

$$
a + bt_3 = \log\left(\frac{k - X_3}{X_3}\right)
$$

Subtracting successive equations gives

$$
b(t_2 - t_1) = \log\left[\frac{(k - X_2)X_1}{(k - X_1)X_2}\right]
$$

$$
b(t_3 - t_2) = \log\left[\frac{(k - X_3)X_2}{(k - X_2)X_3}\right]
$$

Since $t_3 - t_2 = t_2 - t_1$, the left-hand sides are equal, and therefore

$$
\frac{(k - X_3)X_2}{(k - X_2)X_3} = \frac{(k - X_2)X_1}{(k - X_1)X_2}
$$

This equation is solved for $k$. The value of $k$ is then substituted into the difference equations to obtain $b$, and $b$ and $k$ are substituted into the original equations to obtain $a$.

2.5 Elimination of trend

Once a trend curve has been fitted, the trend values $\hat{X}_t$ are computed at each observed time point. The detrended series is the set of residuals

$$
X_t - \hat{X}_t
$$

for the additive model. The residuals contain the seasonal, cyclic and irregular components that remain after the trend has been removed. In the multiplicative model the corresponding operation is division, $X_t / \hat{X}_t$.