Time series analysis: moving averages, exponential smoothing, and seasonal and irregular component

Selection of m

*Very subjective. -Only strict rule is that if there is a seasonal component in the time series, m should equal to the length of seasonality, s. *Example of quarterly time series would give m=4. **Otherwise, the stronger smoothing desired, the higher m that should be selected.

Difference between trend functions and smoothing methods

-A trend function has estimated parameters that remain the same for the whole time series. -Smoothing methods do not assume stability of trend parameters in time. *Assumes that estimated values of the trend component are adapting to the changes in development of time series.

Disadvantages of moving averages

-Arithmetic means are very responsive to extreme values (outliers). -Selection of m is sometimes subjective. -Method cannot be used for forecasting the future development of a time series. -Several values are lost at the beginning and end of time series (due to beginning to calculate averages somewhere in the middle).

Finding p

Simply divide from the m=2p formula (provided you know m). *p and t in formula indicates where to begin calculating mean and where to stop.

Seasonally adjusted data

Time series with a removed seasonal component. *For seasonal differences, respective seasonal different is subtracted from the time series: y(t) - S hat (t). *For seasonal indices, each value of the time series is divided by the corresponding seasonal index: y(t) / S hat (t).

Calculating estimates in regression method with dummy variables

Use the formula y(t)= T(t) + S(t) *T(t) and S(t) will change based on the quarter. **Therefore, a recalculation of these variables must be done before prediction calculation. ***Always consider what quarter number (1-4) and year number in calculations. *Ultimately, the final model of this method is simply the drawn out version of these two components. The symbols are used to simplify.

Simple moving average

Used for when the window's length m is an odd number. (m=2p + 1) *Takes the simple arithmetic mean of p values before time t and y(t) and of the p values after time t. **See the y bar (t) formula. ***Sum divided by m.

Weighted moving average

Used for when the window's length m is either odd or even. *Takes the weighted arithmetic mean of several values.

Centered moving average

Used for when the window's length m is even. (m=2p) *Takes the simple arithmetic mean of p values before time t and y(t) and of the p values after time t. **See the y bar (t) formula. ***1 divided by 2m.

The trend component in regression method with dummy variables incorporates

a bar.

t variable is equal to

a number such as 1, 2, 3, etc. *NOT the year itself. **Therefore, always number off years under analysis beginning with 1 and going forward.

Exponential smoothing determines

a smoothed value Y(t) of a time series y(t) by computing an exponentially weighted average of the current value and all previous values of the time series. =alpha * y(t) + (1-alpha) * Y(t-1) **Only theory of this will be on final exam.

The simple moving average assumes that

all values have the same weights. m=5: 1/5[1, 1, 1, 1, 1]

Usually the series used in regression approach to seasonal component are going to be

based in four quarters.

As with most all regression models used in this course, the regression model used in the regression approach to seasonal component is

linear in parameters. *Meaning that the parameters (Beta, alpha, etc) may be estimated by MLS.

In order to model a trend and seasonal component together, one must use a

regression approach. *Use the multiple linear regression model. y(t) is dependent variable. t, x(1t), x(2t),... are explanatory variables. a(1), a(2),... are the actual seasonal components. x(1t), x(2t),... are dummy variables. **Indicate seasons/quarters and may have values 0 or 1 only.

Smoothing (adaptive) methods attempt to

remove the effect of irregular (random) variations from the values of a time series. *Examples are moving averages and exponential smoothing.

Seasonal components may be measured by

seasonal differences or seasonal indices.

Estimation of the trend component in moving averages is done by

separating the function into short segments ("windows") of the time series by some type of the mean (average). *Move forward with the window through the time series by excluding the first value and adding the next one.

The goal of exponential smoothing is

short-term forecasting. *The smoothed value for time t is prediction (forecast) for the next period (t+1).

Moving averages basically consist of

the replacement of values of time series by averages (means) calculated from them. *The result is a "smoothed" time series (estimate of the trend component). **Can be simple, weighted, or centered.

The higher the m

the smoother the time series.

Aim of seasonal component analysis is

to quantify seasonal variations in a time series.

Exponential trend function

y(t) = B(0) * B(1)^t 1. Calculate ln(y(t)). 2. Do regression analysis. 3. Un-logarithmize resulting intercepts. *=Exp() 4. Insert into above formula.

Seasonal differences

y(t)= T(t) + S(t) + E(t) *Assumes additive decomposition and, therefore, seasonal variations are stable in time and thus compensate themselves (each year). **Measures the effects of different seasons on a time series in ABSOLUTE terms (they are in the SAME UNITS as the time series y(t)). ***Represented as S hat (t).

Seasonal indices

y(t)=T(t)*S(t)*E(t) *Assumes multiplicative decomposition and, therefore, seasonal variations are NOT stable in time and thus change proportionally to the trend component. **Measures the effects of different seasons on a time series in RELATIVE terms (there are DIFFERENT units as the time series y(t)). ***Represented as S hat (t).

Range of seasonal indices

Above-average if > 1 Below-average if < 1

Range of seasonal differences

Above-average if difference > 0 Below-average if difference < 0

Calculating a bar

Add up all three dummy variable coefficients and divide by 4. *In formula, dummy variable coefficients are given as a1, a2, and a3.

Seasonality

Characteristic of a time series in which data experiences regular and predictable changes that rear every calendar year/quarter. *Always a fixed and known period. **Occurs in a seasonal time series, also known as a periodic time series.

Trend component modeling

Constant trend: T(t)=B(0) Linear trend: T(t)=B(0) + B(1)t Quadratic trend: T(t)=B(0) + B(1)t + B(2)t^2

Interpretation of the simple moving average

Estimate of trend component in time period of focus.

Interpretation of a bar

In the period from 1st to 4th quarter, the number of _______ in the first quarter was, on average, by (alpha) below the linear trend.

Durbin-Watson test

Null hypothesis: p(1)=0 "No autocorrelation between e(t) and e(t-1)." Alternative hypothesis: p(1) does NOT equal to 0 "Autocorrelation between e(t) and e(t-1)." **If test statistic is close to 2, we do NOT reject null hypothesis. **If test statistic is close to 0 or 4, we DO reject null hypothesis.

Coefficients for dummy variables on paper and in Excel

Paper: a1 a2 a3 Excel: D1 D2 D3

Forecasting

Prediction of future observations of a time series. *Use T hat formulas for forecasting (future values in time series). **Always remember that conditions can change (no forecast is 100%)-exponential smoothing can help in making more accurate though.

Irregular component

-Estimated by residuals: e(t)= y(t) - T hat (t). *"White noise" properties: -Zero expected value -Constant variance -Mutually uncorrelated -Normally distributed *Durbin-Watson (hypothesis) test is used to verify that e(t) are uncorrelated.

Reasons for studying seasonal component (S(t))

-Yields knowledge useful in short-term forecasting. -May be calculated (estimated) for the purpose of eliminating it from the time series (thereby allowing T(t) to be more easily seen). **Usually quarterly or monthly.

Understanding the idea that dummy variables can be only 0 or 1

1, in the case that the value (a2, a3, etc) that the dummy variable is multiplied by is in the season. 0, in the case that the value (a2, a3, etc) that the dummy variable is multiplied by is NOT in the season.

Steps to regression approach to seasonal component

1. Estimation of regression parameters (usually given). 2. Calculation of the a values' arithmetic mean. 3. Calculation of the seasonal difference. *For the first three seasons: S(i) = a(i) - a bar. *For the last season: S(4) = -a bar. (As the fourth value is equal to 0).

Smoothing constant

=alpha. *Greater than 0 and less than 1. **If close to 1, past values are extremely important and the series decreases quickly.

Length of a moving segment (window's length)

=m *For an odd number of values, m=2p + 1 *For an even number of values, m=2p p=1, 2, 3, ...