Table of Contents
Executive Summary 3
1.Introduction 4
2.Consumer Price Index 4
3.Time Series Analysis 6
3.1 Purpose of Time Series Analysis 6
3.2 Definition of Autoregression Analysis 7
3.3 Methodology 7
4.Results 8
List of Works Cited 10
Appendix 11
Executive Summary
Inflation is one of the main economic indicators of a country measured by the Federal Government using the Consumer Price Index for all urban consumers (CPI-U), published by the U.S. Bureau of Labor Statistics. Calculations are based on annual changes in the Consumer Price Index. Figures are published monthly and annual series are available back to 1860.
The changes in the rate of inflation have a substantial impact on the economic conditions and policies of a country. Therefore, predicting this rate or even trying to regulate and control it, will greatly help to avoid fluctuations of the economy.
One of the ways of trying to find the relationships between the changes of inflation rate over the past years, is by implementing the Autoregression analysis to finding how well does the inflation over the certain period in the past does predict inflation over the future period of time.
The following report concentrates on finding the long-term trends in changes of the US inflation rate in the "Post War Era" period from 1947 to 1997. The authors tried to elicit how well do the previous years’ inflation rates can predict the following years’ rates using the linear Autoregression analysis. The CPI-U indexes and their annual changes have been listed from 1947 to 1997 in Table 1. Using Excel command =RSQ (array Y, array X), R square values for simple linear regression functions were calculated; X ranges from 1 to 20 years, Y from 1 to 30 years. The results are shown in Table 3. The full regression analysis with charts has been run for the five highest regressions. Moreover, the three-dimensional overview of the results of the Autoregression analysis is shown in Chart 2. The sorted list of R squares from the highest to the lowest is shown in Table 4. The regression R squares of inflation rates of 20 years back to 15 years ahead are the highest, almost 99 percent. There is a certain trend of rising R square values within that period of time.
The following report investigates relationships between the changes of inflation rate, by implementing a Time Series Analysis to find how well does the inflation over the certain period in the past does predict inflation over the future period of time. The Time Series Analysis is based on the simple linear Autoregression analysis.
2. CONSUMER PRICE INDEX
The consumer price index (CPI) is the name typically applied in both the United States and other countries to the statistics that measures the price changes of the vast number of goods and services purchased by households. The broad concept underlying the CPI is that it measures the purchasing power of money with respect to a fixed market basket of consumer goods and services.
The CPI for the United States was first published in 1919 and contained data going back to 1860. Indexes for food in some US cities are available prior to that. The CPI is compiled and released monthly by the Bureau of Labor Statistics. The population covered by the consumer price index varies from country to country. In the United States, indexes are calculated for two consumer groups, all-urban consumers and urban wage earners and clerical workers.
CPIs have several important uses. Early in this century, changes in economic conditions required new data related to industrialization, in addition to data already collected for agricultural and external trade. Thus, in the United States, the need arose for a CPI to be used in labor management negotiations, especially inflation accelerated during World War 1.
As economic conditions changed, the use and the design of CPIs changed. During the 1930’s, CPIs were used to measure real income flows of families. The CPI was being used as a deflator, but during a time of price decline it actually inflated incomes. With the advent of Keynesian economics and government short-run policies to stimulate employment and stabilize prices, the CPI grew in importance as an input for policy making. In recent years when worldwide inflation has accelerated, the CPI has become a closely watched statistic. It is widely used to compensate for inflation. Such uses include adjustments to wage contracts, social security and other transfer payments, and income tax brackets.
In the United States and many other countries the statistical agencies make data available on both nonadjusted and seasonally adjusted bases. The former estimates are usually used in the escalation of the contracts, the latter by policymakers in assessing the short-run course of inflation. (Hamburg)
The following report concentrates on finding long-term trends in the annual inflation rate of the US only in the "Post War Era" from 1947 to 1997. Most economists prefer to use this period for projection and analysis into the future.
Table 1 lists the raw CPI-U’s that were used for compiling Chart 1. The overall average rate over the entire 50 year period from 1947 to1997 is about 4.3 percent. The highest annual inflation rate is 14.4 percent in 1947 and the lowest only two years later in 1949 1.2 percent.
Chart 1 illustrates the basic raw data from 1947 to 1997. Violent swings and high levels seem to have leveled out since 1982.
3. Time Series Analysis
Time series analysis is a tool of forecasting, which analyses the data classified by years, quarters, months, or other periodic intervals. Analysis of time series data reveals patterns of growth and change that often can be averaged or measured in such a way that a projection into the future can be made. Both long-range and short-range projections can be used for determining what kind of level business activity will have in the future.
3.1 Purpose of Time Series Analysis
The purpose of the time series analysis is to discover past patterns of growth and change that can be used to predict future patterns. Time series analysis does not provide the accurate data on future trends, but it helps to reduce forecast errors.
All the day everybody is making decisions. Knowingly or unknowingly, consciously or unconsciously, these decisions are based on predictions, and these predictions are usually based on what has happened in the past. For example, when a student goes to class, he predicts that the instructor is already there. His prediction is based on his past experience. The student’s actions are based on the forecast, which in turn is based on past events.
Similarly, this report follows the basic research question: "Is inflation over the past X years a good predictor of inflation over the subsequent Y years?"
3.2 Definition of Autoregression Analysis
The Autoregression analysis is estimation of the value of a random variable given that the value of an associated variable is known. Simple Regression Analysis indicates that the value of a dependent variable is estimated on the basis of one independent variable. The linear model that represents the simple linear regression is:
In this equation ‘a’ represents the point of intersection of the trend line with the Y-axis, whereas ‘b’ represents the slope of the trend line.
In the case of an Autoregression analysis the regression of Yt is done against values of itself in earlier periods (that is, Yt-1 or Yt-2 or…). The earlier periods are called "lagged variables". To give an example based on the annual change of the CPI-U: a one period lag would use last year’s changes in the CPI-U to predict this year’s changes in the CPI-U; a two period lag would use the data from the change two years ago to predict today’s change.
3.3 Methodology
The simple linear Autoregression analysis was used for determining how do the previous years’ inflation rates influence the following years’ rates, basing on the linear model that represents the simple linear regression (Y=a+bX), X represents the compounded rate of inflation in certain previous periods of time and Y represents the compounded inflation rate in certain subsequent periods of time.
For calculating the R Square values, the CPI-U indexes were listed from 1946 to1997. Then the percentage change from one CPI-U index to the next one was calculated (see Table 1). For achieving the highest calculation accuracy of the R square values, the compounded rate of inflation from one year lagged data up to 30 years lagged data (see Table2) was calculated.
The Autoregression analysis was calculated by the Excel command =RSQ (array Y, array X), which is based on the unadjusted R Square value for simple linear regression analysis. In a matrix table the unadjusted R Square values were calculated from one year up to 20 years back and similarly one year up to 30 years ahead. The general rule to calculate the R Square value is to use at least 30 data-points per variable, but in this paper, a minimum of 12 data-points per variable were considered for further investigation at a later time. Therefore, the R Square values in the lower right corner of Table 3 have not been calculated. The results of the calculation of the R Square values are listed in Table 3. Table 4 provides the reader with the sorted overview of the results. Furthermore, Chart 2 gives a three-dimensional overview of the results of the Autoregression analysis.
4. Results
The full regressions with charts for the five highest R squares have been run. After analyzing the results, we found that residual plots of 3 of the five highest R squares are not random. However, for determining the reason of this pattern, further analysis is required. From the results shown in the three-dimensional overview (Chart 2) and sorted list of R squares (Table 4), we can see that R squares of 20 years back to 15 years ahead correlations of changes in CPI-U are very high, approximately 99 percent (the highest is 98.95 percent). We could say that it is almost perfect correlation and we have found the clue on how to predict inflation in the future. However, even though we have found very high R square, we doubt to use this outcome as a tool for predicting inflation rate over the next years. Trying to tie this paradox to major economic shifts or changes in the US economy, we couldn’t find certain clue to explain that puzzle. The reason of this extraordinary result is over the scope of this paper and needs further and more detailed statistical analysis.
list of works cited
Hamburg, Morris. Basic Statistics: a modern approach. 1979 .
Huxley, Stephen. Huxley Lecture Notes for Data Analysis. 1997.
Kazmier, Leonard J. Business Statistics. 1995.
Appendix
To view accompanying charts, click here