DHSY start their analysis with a comprehensive account of the properties of consumption and income series via various kinds of graphs. DHSY describe the data they have used, and analyze the data by various types of Graphs and correlogram. The data file used by DHSY is available in the following link.
The two series DHSY use are:
a. Personal disposable income Yt
b. Consumer expenditure on non-durable goods Ct
They make following kinds of graph of the data:
1. Line graph of the two series Ct and Yt
2. Line graph of the quarterly difference of the two series Δ4Ct and Δ4Yt
3. Graph of average propensity to consume Ct /Yt
4. Scatter plot of series Ct and Yt
5. Scatter Plot of the quarterly difference of the two series Δ4Ct and Δ4Yt
6. Line graph of log of Ct
They discuss each graph separately. The first graph i.e. line graph of the two series Ct and Yt
The seasonal pattern is very clearly visible because of which the graph makes ‘hills’. This seasonal pattern is visible in both consumption and income.
The graph shows following:
a. Income series show instability in the seasonal pattern (size of ‘hills’). For example in 1966-1970, the seasonality becomes less prominent.
b. The graph of Ct shows stability in the seasonality of data.
c. Seasonal pattern becomes more wide or ‘elongated’ in the era 1971-76.
In graph 2, DHSY plot line graph of the quarterly difference of the two series Δ4Ct and Δ4Yt and this graph is reproduced below:
is
The Graph 2 shows that:
a. There is no seasonality in the graph, there are no ‘hills’ at regular interval of four periods
b. The quarterly difference of consumption series is smoother than the income series. The variation in income series larger than that of consumption series.
c. The variance f the period 1971-76 is more than that of for the early period. This is true for both income and consumption series
Graph 3 produced by DHSY is the graph of average propensity to consume. The APC is calculated by dividing consumption on income i.e. APC= Ct /Yt . The graph of this series is reproduced as follows:
One can see from this graph that:
a. APC has observed a continuous decline if we ignore the seasonality
b. The average APC in the beginning was about 0.9 which decline to about 0.8 by the end of sample period
Graph 4 produced by DHSY is scatter plot of Ct and Yt which is reproduced as sunder:
Fifth graph DHSY produce is the scatter plot of Δ4Ct and Δ4Yt. This graph is given as under:
The graph 4 and five give roughly similar message except that the seasonal pattern vanishes in the graph of seasonally differenced series.
Lastly, DHSY produce the graph of logarithm of Consumption series which is as under:
This graph shows that the magnitude of seasonal pattern is roughly smooth after taking the log transform.
Meanwhile DHSY mention a paper by Prothero and Wallis (1977) who obtained univaraite ARMA models for the consumption and Income series. DHSY mention that they found following ARMA models for consumption and income series for sample period 1957-1967.
At the end DHSY provide the correlogram of the Ct, Yt, Δ4Ct and Δ4Yt.
The correlogram is reproduced below as:
One can see from this correlogram that:
a. The correlation of Ct and Yt with their lags are stronger than that of Δ4Ct and Δ4Yt
b. Correlation for lag 4,8,12 are stronger than other lags indicating high seasonality in case of Ct and Yt . No such correlation is observed for Δ4Ct and Δ4Yt