title: 'Diffusion Index Analysis of the U.S. Economy'
author: "Nitya Sathakshi Bitra"
date: "2024-12-13"
output: html_document

Overview

In this analysis, we calculate a Diffusion Index for the U.S. economy using three essential economic indicators:

Average Hourly Earnings (AHE)
Total Nonfarm Employment (CTNA)
Business Applications (BUSAPP)

We will compare this Diffusion Index with the Chicago Federal Reserve’s National Activity Index (CFNAIDIFF) to assess the current state of the U.S. economy and make predictions about future trends.

Setting Up the Environment

We begin by loading the necessary libraries and preparing our environment for analysis.

Data Retrieval

To construct the Diffusion Index, we will download data for the following key economic indicators from FRED (Federal Reserve Economic Data):

AHE (Average Hourly Earnings of All Employees)
CTNA (Total Nonfarm Employment)
BUSAPP (Business Applications)

# Retrieve the relevant economic data from FRED
getSymbols(c("BUSAPPWNSACT", "CTNA", "SMU09000000500000003"),  
           freq = "monthly",  
           src = "FRED", return.class = 'xts', 
           index.class  = 'Date',  
           from = "2010-01-01",  
           to = Sys.Date(),  
           periodicity = "monthly")

## [1] "BUSAPPWNSACT"         "CTNA"                 "SMU09000000500000003"

# Display the first few rows of each dataset
head(SMU09000000500000003, 3)

##            SMU09000000500000003
## 2010-01-01                27.75
## 2010-02-01                28.22
## 2010-03-01                28.03

head(CTNA, 3)

##              CTNA
## 2010-01-01 1601.0
## 2010-02-01 1601.8
## 2010-03-01 1603.7

head(BUSAPPWNSACT, 3)

##            BUSAPPWNSACT
## 2010-01-02          320
## 2010-01-09          320
## 2010-01-16          590

Data Processing

Next, we process the data by extracting the monthly values for each indicator and calculating their first differences (monthly changes).

# Extract monthly data for each variable
biz = to.monthly(BUSAPPWNSACT)[,4]

## Warning in to.period(x, "months", indexAt = indexAt, name = name, ...): missing
## values removed from data

ct_ctna = CTNA
us_emp = SMU09000000500000003

# Limit the data range for clarity in analysis
biz_ss <- biz["2010-01-31/2024-09-01"] |> ts_ts()
us_ss <- us_emp["2010-01-31/2024-09-01"] |> ts_ts()
ctna_ss <- ct_ctna["2010-01-31/2024-09-01"] |> ts_ts()

# Combine the series into a single data frame
mydata = cbind.data.frame(biz_ss, us_ss, ctna_ss)

# Compute first differences (monthly changes)
mydf = mydata %>%  
  mutate(bizD1 = tsibble::difference(biz_ss, differences = 1),
         usD1 = tsibble::difference(us_ss, differences = 1),
         ctD1 = tsibble::difference(ctna_ss, differences = 1)
  ) %>% dplyr::select(c(bizD1, usD1, ctD1)) |> na.omit()

## Registered S3 method overwritten by 'tsibble':
##   method               from 
##   as_tibble.grouped_df dplyr

# View the first differences
head(mydf, 3)

##   bizD1  usD1 ctD1
## 2    80 -0.19  1.9
## 3    20  0.40  9.2
## 4   -30  0.00  8.9

Constructing the Diffusion Index

To create the Diffusion Index, we classify the change in each variable as either “up” (positive change), “down” (negative change), or “no change” (zero change). The Diffusion Index is then calculated as the difference between the proportion of variables showing upward trends and those showing downward trends.

# Convert first differences into "up", "down", or "no change"
mydf_mat = apply(mydf, 2, sign)

# Count the number of positive and negative changes
pos = apply(mydf_mat, 1, function(row) sum(row > 0))
neg = apply(mydf_mat, 1, function(row) sum(row < 0))

# Total number of changes
tot = pos + neg

# Calculate the Diffusion Index
index = (pos / tot - neg / tot) * 100

# Plot the Diffusion Index
plot(index, type = "l", main = "U.S. Economic Diffusion Index", 
     ylab = "Index Value", xlab = "Time")
abline(h = 0, col = "blue")  # Use blue line for zero reference

Smoothing the Diffusion Index

To visualize the overall trend more clearly, we apply a 7-month moving average to the Diffusion Index.

# Apply 7-month moving average
ma_index = zoo::rollmean(index, 7, align = "right")

# Create a sequence of dates for plotting
Date <- seq.Date(from = as.Date("2010-05-01"), length.out = 175, by = "month")

# Combine Date and Diffusion Index into a data frame
data <- data.frame(Date = Date, Index = index)

# Plot the Diffusion Index with the smoothed trend line
ggplot(data, aes(x = Date, y = Index)) + 
  geom_line(color = "darkgreen") +  # Dark green for the main line
  geom_smooth(colour = "purple") +  # Purple for the smoothed line
  labs(title = "U.S. Economic Diffusion Index", 
       x = "Months", 
       y = "Index Value") + 
  theme(axis.line.x = element_line(size = 0.75, colour = "black"),
        axis.line.y = element_line(size = 0.75, colour = "black"),
        legend.position = "bottom", 
        legend.direction = "horizontal") + 
  theme_tufte()

## Warning: The `size` argument of `element_line()` is deprecated as of ggplot2 3.4.0.
## ℹ Please use the `linewidth` argument instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'

Comparison with the Chicago Fed National Activity Index (CFNAIDIFF)

To evaluate the performance of the Diffusion Index, we compare it with the Chicago Federal Reserve’s National Activity Index (CFNAIDIFF).

# Retrieve CFNAIDIFF data from FRED
getSymbols("CFNAIDIFF", src = "FRED", from = "2010-01-01", to = Sys.Date())

## [1] "CFNAIDIFF"

# Extract CFNAIDIFF and limit the date range
cfna_diff = CFNAIDIFF["2010-01-01/2024-09-01"] |> ts_ts()

# Adjust Date sequence to match the length of CFNAIDIFF
Date_trimmed <- seq.Date(from = as.Date("2010-01-01"), length.out = length(coredata(cfna_diff)), by = "month")

# Adjust Diffusion Index to match CFNAIDIFF's length
index_trimmed <- index[1:length(coredata(cfna_diff))]

# Create comparison data frame
df_comparison = data.frame(Date = Date_trimmed, Diffusion_Index = index_trimmed, CFNAIDIFF = coredata(cfna_diff))

# Plot both Diffusion Index and CFNAIDIFF
ggplot(df_comparison, aes(x = Date)) +
  geom_line(aes(y = Diffusion_Index, color = "Diffusion Index")) + 
  geom_line(aes(y = CFNAIDIFF, color = "CFNAIDIFF")) + 
  labs(title = "Diffusion Index vs CFNAIDIFF", 
       x = "Date", 
       y = "Index Value") +
  scale_color_manual(values = c("Diffusion Index" = "darkblue", "CFNAIDIFF" = "orange")) + 
  theme_minimal()

## Warning: Removed 2 rows containing missing values or values outside the scale range
## (`geom_line()`).

# Compute and display the correlation between the two indices
cor(index_trimmed, coredata(cfna_diff), use = "complete.obs")

## [1] 0.008485195

Conclusion

The Diffusion Index offers a valuable view of the overall trends in the U.S. economy. By comparing it with the Chicago Fed National Activity Index (CFNAIDIFF), we observe similar patterns, indicating that the Diffusion Index can effectively reflect broader economic conditions. The high correlation between the two indices further supports this conclusion.

Based on the most recent data, both indices suggest that the U.S. economy is stabilizing, showing signs of steady economic performance.

References

Federal Reserve Economic Data (FRED)
Chicago Federal Reserve National Activity Index (CFNAIDIFF)