Time series are particularly useful to track variables such as revenues, costs, and profits over time. Time series models help evaluate performance and make predictions. Consider the following and respond in a minimum of 175 words:

• Time series decomposition seeks to separate the time series (Y) into 4 components: trend (T), cycle (C), seasonal (S), and irregular (I). What is the difference between these components?
• The model can be additive or multiplicative.When we do use an additive model? When do we use a multiplicative model?
• The following list gives the gross federal debt(in millions of dollars) for the U.S. every 5 years from 1945 to 2000:

Year    Gross Federal Debt ($millions)

1945    260,123

1950    256,853

1955    274,366

1960    290,525

1965    322,318

1970    380,921

1975    541,925

1980    909,050

1985    1,817,521

1990    3,206,564

1995    4,921,005

2000    5,686,338

• Construct a scatter plot with this data. Do you observe a trend? If so, what type of trend do you observe?
• Use Excel to fit a linear trend and an exponential trend to the data. Display the models and their respective r^2.
• Interpret both models. Which model seems to be more appropriate? Why?

ANSWER

Week 6 Discussion

Time series decomposition is a forecasting method and technique that identifies two separate components in the underlying basic pattern existing in a data series characterizing the economic or business series, smoothening past data, and distinguishing the pattern in a data series from randomness. Components of time series decomposition are trend-cycle factors, representing the long-term changes in the data series, and the seasonal-irregular factors periodic fluctuations caused by unknown factors, periodically or in the short-term. This model assumes that in every data series, there is a pattern and error present (Data = Pattern + Error// Y_t =ƒ (S_t, T_t, E_t) (ABS, n.d.). Therefore, the model’s trend-cycle is a long-term movement or direction in a time series without any irregular effects. The seasonal component is systemic, calendar-related movements consisting of stable periodic impacts that occur with respect to time, such as the weather fluctuations that arise in relation to their seasons (timing). The irregular component consists of the unsystematic and unpredictable short-term fluctuations that occur after estimating the trend and seasonal components in a time series.

Time series decomposition can be used in two approaches: additive (Y_t = S_t, +T_t + E_t) and multiplicative models (Y_t = S_t, x T_t x E_t). An additive model is used when the seasonal fluctuation and irregular variations magnitude do not vary or change with the series levels (when the trend rises or falls). On the other hand, the multiplicative model is used when the amplitude or magnitude of the seasonal fluctuations and irregular variations increase with the series level (as the trend rises) (ABS, n.d.).

The scatter plot shows a positive trend as the Gross Federal Debt increases with the increasing years.

Liner trend

Exponential trend

The r^2 in the linear trend is R² =0.7202, and that of the exponential trend is R² =0.8913. This shows that the exponential trend is the most useful being the highest, and shows the variations and fluctuations as data values of the GFD rise and fall at higher rates, instead of the steady pace expressed by the linear trend.

Reference

ABS. (n.d.). Time Series Analysis: The Basics. Retrieved from Australian Bureau of Statistics: https://www.abs.gov.au/websitedbs/d3310114.nsf/home/time+series+analysis:+the+basics