ECON 102​ ​ ​ ​ ​ ​ ​ ​ ​ Assignment 1

Part A

Question 1: Plot the series using a line chart. Briefly describe what you see: Is it a positive or negative
trend? Is the trend increasing? What kind of short term fluctuations do you observe?

The graph above shows a clear positive trend on average that is generally increasing over the years of
1950-2020. The trend is definitely not linear as there are many short-term fluctuations between the time
horizon of the data. We can observe rapid ups and downs in the expenditure over time and these
short-term fluctuations are consistent over 1950-2020. This may be due to reduced spending in certain
quarters and increased spending in subsequent quarters, but the overall trend is positive and increasing
over time regardless of the fluctuations. Furthermore, the slope on average is relatively consistent as we
can see the graph growing at a steady pace without any shifts in the steepness of the slope. This could
mean that expenditure is growing at a steady pace with short-term fluctuations in between quarters.
Question 2: Answer the previous question using the log-scale. Can you tell if the growth rate is
increasing or decreasing on average over the period?

Using the log-scale to calculate the expenditure overtime, we can see that similar to the first chart, there is
a clear positive trend and the overall expenditure is increasing over 1950-2020. From roughly 1950 to
1980, it appears that the graph is increasing at a steeper rate as compared to 1980-2020, where the slope
seems to be a bit flatter compared to the period before. This could mean that the trend of expenditure is on
a gradual decrease over time but the overall trend still remains positive. There are regular short-term
fluctuations in the trend throughout the time horizon (similar to the previous graph), which could have
been caused by short-term cyclical variations between quarters. Overall, since the graph’s slope is slightly
steeper from 1950-1980 and a bit flatter from 1980-2020, we can assume that the growth rate may be
decreasing on average from the time horizon of 1950-2020, despite the overall trend of the graph being
clearly positive.
Question 3: To better see how the growth rate evolves through time, plot the annualized growth rate of
consumption expenditure. Describe what you see. Is it constant on average?

The annualized growth rate of consumption expenditure fluctuates significantly throughout the time
horizon of 1950-2020. From 1950-1970, we can see that there are some significantly high peaks of nearly
100% or more and troughs that are at or below -50%. This shows the high volatility in consumption
expenditure in the early stages of the time horizon. However, as we look beyond the 1970 mark up until
2020, there appears to be less functions as compared to the period before. The volatility is still present as
the peaks extend to nearly 50% or more and the troughs are at hitting points close to -30% to -50%.
Despite that, the reduction in volatility is present later on in the time horizon which could signify that
growth rates are most consistent and predictable. However, looking at the overall time horizon, the
annualized growth rates are not constant on average as there is high average variability throughout the
time horizon of 1950-2020.
Part B

Question 1: Fit a linear and quadratic trends to your series. Then, create a line chart with your original
series and the two trends. Which trend seems to best fit the series? Explain.

The quadratic trend (dotted red line) best represents the overall series. When we look at the linear trend
(dotted blue line), it shows a constant average increase over time from 1950-2020. However, that does not
fit the series accurately as consumption expenditure is non-linear. This can be proven by looking at the
linear trend relative to the expenditure series. The linear trend is on top of the expenditure growth initially
(1950s), at the bottom during the middle (1980s), and at the top again in the end (2020). This is an
inaccurate description of the overall trend. Looking at the quadratic trend, we can see that it generally
follows the shape of the expenditure data so it is a better representation of the data points in the series
compared to the linear series. The quadratic trend adjusts to the growth over the time horizon and
accurately follows the original series, leading to the conclusion that the quadratic trend is a better
representation of the series compared to linear trend.
Question 2: Fit a linear and quadratic trends to the log of your series. Then, create a line chart with the
log of your series and the two trends. Which trend seems to best fit the series? Do you see a difference
between the best trend in this question and in the previous one? Explain.

Similar to the finding in question 1, we can see that the quadratic trend (dotted red line) fits the log series
more accurately compared to the linear trend (dotted blue line). The slight curvature in the log series is
captured quite well in the quadratic trend as it follows it very closely. The linear trend, similar to the
findings above, hovers above and below the log series without accurately representing the entire series as
a whole. For instance, it starts (1950s) and ends (2020) at the higher end of the series and falls to the
lower end during the middle (1980s). Due to very similar conclusions being drawn in both the log series
and the series above, there is no difference between the best trend in this question and the previous one.
Question 3: Plot the detrended series using the trend that best fits the series. Briefly describe what you
see: Do you better detect short term fluctuations?

Short-term fluctuations become much more apparent in the detrended log of consumption expenditure
series using the quadratic trend. From 1950 to 1955, there were a lot of short-term fluctuations as there
were many significant peaks and troughs ranging from 0.1 to -0.14, respectively. As we progress in the
time horizons, the short-term fluctuations slowly become less volatile as the range around the 2020 era is
roughly -0.01 to 0.8. All in all, there are more fluctuations initially compared to later on in the time
horizon and these short-term fluctuations become easier to detect using this graph.
Question 4: Using a moving average of order 5, compute the cyclical component of your series. Then,
plot the cycle and briefly describe what you see: interpret the values of some peaks and troughs.

Generally speaking, the graph above shows the cyclicality within the time horizon between 1950 to 2020
of expenditure using a moving average of order 5. As for analyzing the peaks and troughs:
Peaks Troughs

●​ We can see that from the late 1950s up ●​ In the early 1950s (roughly 1953), we can
until late 1960s (let’s assume 1957-1967), see that the cyclical components reached
the cyclical component reached roughly nearly -0.06. This shows that there was a
0.04 five times. This could symbolize a substantial reduction in expenditures
period of high expenditure as the cyclical around this time.
component is relatively high compared to ●​ Additionally, there was a significant
the rest of the time horizon. trough around 1995, where the cyclical
●​ Furthermore, it peaks again 1996-1997 component reached below -0.03 since the
and 2019, reaching above 0.03. This once mid 1950s. This almost happened again
again represents a period of increased close to the mid 2000s (around 2005 and
consumption expenditure. 2008). This could be due to a number of
reasons, including economic recessions,
the global financial crisis of 2008 and
much more.
Question 5: Plot the low frequency of your series and briefly describe what you see.

As a general overview, the line graph above represents the low frequency log of expenditure over the time
horizon of 1950-2020. The graph shows a clear positive and increasing trend, which represents a gradual
increase in overall consumption expenditure during the time horizon. There are not many short-term
fluctuations and despite there being a few variations in the general slope of the series (some parts being
steeper compared to others; e.g. early 1950s vs. late 1950s), the overall low frequency of log of
consumption expenditure is positive and increasing.
Question 6: Compute the seasonal component and represent it on a bar chart (only the 4 quarters).
Interpret the four seasonal values.

Quarter 1: As seen in the graph, Q1 represents a negative seasonal component with its value being
roughly -0.013. This means that the consumption expenditure during this quarter is below the average
annual consumption expenditure.
Quarter 2: As seen in the graph, Q2 represents a negative seasonal component with its value being
roughly -0.0135. This means that the consumption expenditure during this quarter is below the average
annual consumption expenditure and since the value is lower than Q1, it could also mean that the average
annual consumption expenditure drops further in this quarter.
Quarter 3: As seen in the graph, Q3 represents a significantly positive seasonal component with its value
being roughly 0.07. This means that the consumption expenditure during this quarter is well above the
average annual consumption expenditure as the bar is a lot higher compared to other quarters as well as
the average.
Quarter 4: As seen in the graph, Q4 represents a negative seasonal component with its value being
roughly -0.035. This means that the consumption expenditure during this quarter is significantly below
the average annual consumption expenditure as well as Q1 and Q2, meaning that average consumption
expenditure drops even further in this quarter.
Generally speaking, Q3 is the only quarter in which we experience a positive seasonal component. Q1, Q2
and Q4 are periods where the average consumption expenditure drops between -0.013 to -0.035. The
drops in the first two quarters lead to large expenditures in the third quarter, but that large expenditure
could be the reason that there is a negative seasonal component in Q4.
Part C

Question 1: Create a scatter plot of your series expressed in logs against the selected series also expressed
in logs.

Question 2: Using the log of the selected series, compute its cyclical component. Then create a scatter
plot of this cycle with the cycle of your series computed in Part B.
Question 3: Looking at the two scatter plots, what can you say about the type of comovement between
the two series?
Datasets: dat29 (assigned dataset) and dat1 (chosen dataset)
Scatter Plot of Log Expenditures (Graph 1) Scatter Plot of Cyclical Components (Graph 2)

By observing the correlation of data between By looking at the graph, there does not appear to
dat29 and dat1, we can conclude that there be any trend and the data points are primarily
appears to be a positive linear correlation. This scattered throughout. There is no general direction
means there is positive comovement between the as to where they are going from a longer-term
two datasets. As dat29 (x-axis) increases in value, perspective and for that reason, there is no
so does dat1 (y-axis). This is a primary indicator relationship between the cyclical components of
of positive linear correlation. dat29 and dat1. We are unable to tell what
happens to once variable as the other increases or
decreases due to the scattered distribution,
meaning that there is no comovement between
dat29 and dat1.