The Tech Decoupling

Our data cover a time period starting from March 21, 2006 to March 21, 2023. The tech index in NASDAQ 100 was constituted for the first time on Feb 22, 2006. Because just after the launch of an index, there may be irrational fluctuations in the index. In order to avoid this, we collect data starting from one month after i.e. March 21, 2006. We stop on March 21, 2023 to complete round-off the time period to 17 years. Our data is collected at a daily frequency. Therefore, we have a total of 4297 data points for a given time series. We download data from yahoo-finance website.

We collect the information on two main time-series : technology and non-technology

1. 1.

   Tech: NASDAQ 100 Technology Sector (NDXT): This index reports the equal-weighted index based on the securities of the NASDAQ-100 Index that are classified as Technology according to the Industry Classification Benchmark (ICB) classification system. On February 22, 2006, the NASDAQ-100 Technology Sector Index began with a base value of 1000.00. To know further on how technology sectors are defined, one can refer to the ICB definitions here.

2. 2.

   Non-tech: NASDAQ 100 Ex Tech Sector (NDXX): This index reports an equal-weighted index based on the securities of the NASDAQ-100 Index that are not classified as Technology according to the Industry Classification Benchmark (ICB) classification system. Similar to the tech index, this index too began with a base value of 1000.00 on Feb 22, 2006.

Broadly, our data is already cleaned and processed by the data provider’s website. We hardly have any missing values in our data, therefore no missing value treatment is required. Since the data is taken from the same stock market index of the same country (USA), therefore, the trading days and economic factors are common to both tech and non-tech time series in our case. Therefore, the matching of the same reference unit is not an issue.

In particular, in the R-markdown notebook, we show that our series $y_{t}$ has three missing values out of 4277 and the other series $x_{t}$ does not have any missing value. Therefore, we just drop the three observations as they are not going to make any big difference in our analysis. After this deletion, we are left with 4274 observations or rows in total.

After cleaning the data, we need to process it in $R$ in order to analyze it as a time series object. We mainly use $R$-packages xts and zoo for our data-processing and later for analysis. First of all, we convert the data two time-series into numeric form. Then we read the Date column as date-object in $R$. After this, we convert the two time-series indices stored in two columns of a data-frame object into a time-series object in $R$. Now $R$-software recognizes our two series as time-series object. Now, we can apply the pre-built functions meant for time series objects on our series. Therefore, we are ready to apply functions to do data exploration and deeper analysis. Hence we move to the exploratory data analysis (EDA) section next.

In this section, we briefly summarize the tools used for data exploration. These tools are typically different from the i.i.d. data regime, therefore, I thought that I should give a brief introduction of these tools before using them.

1. 1.

   ACFs: The autocorrelation function (ACF) is a plot of the correlation between a time series and its lagged values. It is a useful tool for understanding the degree of autocorrelation in a time series and can be used to identify the order of an autoregressive (AR) or moving average (MA) model. The ACF plot here is showing lags on the x-axis and correlation coefficients on the y-axis. The correlation coefficients range from -1 to 1, with 1 indicating a perfect positive correlation and -1 indicating a perfect negative correlation. A value of 0 indicates no correlation.

2. 2.

   PACFs: In James Hamilton’s book “Time Series Analysis”, the PACF is introduced in Chapter 2 as a method for determining the order of an autoregressive (AR) model. The PACF is used to identify the number of lags that need to be included in the AR model by examining the magnitude and significance of the partial autocorrelations at each lag. A large partial autocorrelation at a particular lag indicates a strong correlation between the original series and that lagged value, which suggests that the lag should be included in the AR model. Hamilton notes that both ACF and PACF are useful tools for identifying the order of an AR model, but they should not be used in isolation. Both of them needed together to decide on AR and MA lag models.

3. 3.

   Co-integration test: In order to get an idea about the correlation between two series, we do a naive correlation. To comment on the long-run relationship and with technical confidence, I then do Johanson’s co-integration test.

4. 4.

   Comparative Graph: We plot the two series together to see how they behave. This gives us an idea of what exactly different thing happens to the two series at some point in time. I didn’t plot both series individually to avoid repeated grapes and to save some space.

5. 5.

   Summary statistics: We draw a table to present the five-point summary, average value, the overall change from the beginning date to the last date, and variation in the data. This table gives us an idea about the distribution and overall percentage returns of the series.

In this section, we’ll get to know our tech index more in detail. First of all, we want to say that we have data on this index since the time it came into existence in 2006. The base value of the index is 1000. We present a table showing a five-point summary and the average value of $y_{t}$ below in table-1

The starting value means the value of the index when we started collecting data which is the value on March 21, 2006. The latest value of the index (on March 20, 2023) is 6783, which implies that the tech index has increased by around 6-fold in the last 17 years. This is an impressive annual return of around 33% on average. We also see substantial volatility in this series with the standard deviation being almost the same order as the average value.

In order to know more about the time-dependence structure of the series, we plot the autocorrelation function (ACF) and partial auto-correlation function (PACFs) in figure-1. In the case of our series $y_{t}$, the ACFs don’t become zero. To be technically correct, ACFs fall in a confidence interval around zero that is not significantly different from zero after taking approximately 1400 lags into account. However, handling such a high number of lags is neither easy nor parsimonious from the modeling point of view. Since the autocorrelation function decays very very slowly, there may be a unit root component that is making the series non-stationary for practical purposes. Therefore, we need to do a unit-root test for this series before proceeding further. We are not deciding on the AR order of the series at this moment. Doing unit-root tests is beyond the capacity of EDA section, we defer it to the analysis section to be done later on.

In this section, we are going to learn about our non-technology index i.e. our series $x_{t}$ in detail. The non-technology index is formally known as the Ex-Technology sector index at NASDAQ100 and is represented by NDXX as discussed in the data section previously. Below, we present a summary table (Table-2) that gives us a snapshot of how our series has been in the last 17 years of its existence.

We note that the base value of this index is 1000 but we started collecting data one month after, therefore, our starting value means the value of the index on March 21, 2006. We note that the latest value of the index is 4675, which is a 3.55-fold increase from its starting value. If we convert this change into an average annual return, we find that we get around 21% annual returns. We note that this return is lower than the technology index return which was 33% annually. We further note that the volatility of this series is less than $y_{t}$ because the standard deviation here is half of the average value while in the series $y_{t}$, it was around 75% of the mean value.
In order to uncover the time-dependence pattern of this series, we look at the ACFs and PACFs.

The two graphs i.e. ACFs and PACFs of this series (Figure-2) are qualitatively similar to that of series $y_{t}$. We note that the ACFs of this series decay even slower than series $y_{t}$. The PACFs go to zero immediately in the first lag. Therefore, the conclusions are similar to that of what we said about series $y_{t}$. We need to do a unit root test to first check whether the series is stationary or not. Therefore, we defer the remaining analysis for the later sections.

Analyzing the two series together gives very important insights into their relationships. We report two graphs and one test in this section. First of all, we simply plot a graph of two-time series in a single plot (Figure-3).

We can see that the two series closely mingle with each other till around 2016. After that, they started de-coupling. The covid-19 shocks further widened this decoupling of the two series. Too much close co-movement of the two series directly tells us that these series may be closely correlated.
In order to see their cross-correlations, we draw a graph (Figure-4). We keep the maximum lag order equal to 1000, meaning that the cross-correlation is to be calculated between the current value and the 1000th lagged value of the two series.

We note that the correlation coefficient decays very very slowly like the ACFs of individual series. This graph suggests that what happened 1000 trading days ago in the tech index will affect the non-tech index today. This claim looks to be easily believable. There could be some spurious correlation. We need to test the long-run relationship between $x_{t}$ and $y_{t}$ series.

A spurious correlation is a concern when analyzing time series data because many economic and financial variables are highly correlated with each other, even when they do not have a causal relationship. This can be due to the influence of common external factors or to the fact that the variables are non-stationary and exhibit trends that are not related to each other. Co-integration, on the other hand, is an important concept in time series analysis because it allows for the identification of long-term relationships between non-stationary variables. Hamilton emphasizes the importance of testing for co-integration before estimating regression models, as failure to do so can result in spurious regression and incorrect inference.
Hamilton (2020) also discusses various methods for testing for co-integration, including the Engle-Granger two-step procedure and the Johansen procedure. These methods involve testing for the presence of a common stochastic trend between variables and can help to identify whether a meaningful long-run relationship exists between variables. Therefore, we go ahead and do a co-integration test here. Here, we only report (Table-3) the test statistics and critical values which are necessary to take a decision. The rest of the details of the result of Johansen’s test are omitted for the simplicity of the presentation. The Johansen cointegration test is implemented by the ca.jo() function in R. The null hypothesis is:

In other words, the null hypothesis is that there are no linear combinations of the variables that are stationary and share a long-run relationship.

Interpretation of the results: In order to conduct a co-integration test, the two series have to be stationary. But we are not sure in our case whether they are stationary or not. Assuming that both $y_{t}$ and $x_{t}$ are stationary, then this test statistic (=3.6) is less than the critical values at 1%, 5%, and 10% significance levels implying that we cannot reject the null hypothesis. That is, there is no cointegration. Hence, we conclude that there is no evidence of the presence of co-integration or long-run relationship between series $y_{t}$ and series $x_{t}$. Since there is no co-integration, we cannot say that there is a long-run relationship between our two series. But the cross-correlation graphs tell the opposite story. However all this is under the assumption that both of our series is stationary, since we haven’t tested this assumption, our conclusion may be wrong. Therefore, we need to do a detailed stationarity analysis before concluding anything.

In this section we got to know our time series better. We now know how they have behaved in the last 17 years in terms of volatility, and distributions. We also looked at the cross-relationship between the two variables. Our main takeaway is that there is a strong possibility that both time series are non-stationary. Using cross-correlation and co-integration tests, we further strengthen the requirement of stationarity analysis. Note that the results of our co-integration test may not be valid because we don’t know whether $x_{t}$ and $y_{t}$ are co-integrated or not.

In simple words, a stationary time series is a series whose statistical properties such as mean, variance, and co-variance remain the same over time. There is more than one notion of stationarity, like strong stationarity, weak or covariance stationarity etc. However, weak stationarity is the main notion used in the literature. We define it in the following text. We call a time series $y_{t}$ to be weak or covariance stationary if the following holds:

1. 1.

   $\mathbb{E}(y_{t})=\mu$ : The mean of the series doesn’t depend on time.

2. 2.

   $Var(y_{t})=\sigma^{2}_{y}<\infty\quad$: Variance of the series is finite and doesn’t depend on time.

3. 3.

   $Cov(y_{t},y_{t-s})=\gamma(s)\quad$: Co-variance between any two-time terms depends only on the time difference between the point and not on the absolute time.

So basically the three properties are not a function of time. In other words, we can say that $\mathbb{E}(y_{t})=\mathbb{E}(y_{t+i})=\mu$ for all $i$. Also, $Cov(y_{1},y_{2})=Cov(y_{2},y_{3})=Cov(y_{99},y_{100})$.

Having defined the stationarity, I now discuss, why is this a good property and why care about it. There are several reasons but I want to highlight three main reasons. First, stationarity makes the behavior of the series consistent and predictable, making it easier to model and analyze. Second, is that many of the standard time series models, such as autoregressive integrated moving average (ARIMA) models, assume stationarity. These models are widely used in forecasting and other applications, so it is important to ensure that the underlying data is stationary before applying them. The third reason why stationarity is important is that non-stationary time series can exhibit spurious correlations, which can lead to misleading conclusions. For example, if two non-stationary time series appear to be highly correlated, this may simply be due to the fact that both series are trending upwards over time, rather than a genuine relationship between the two series.

Consider the following simple AR(1) time series model:

The coefficient $\beta_{1}$ represents the marginal effect of $y_{t-1}$ on $y_{t}$. If $|\beta_{1}|<1$, then the effect of lag term on the current time series dies out with time, and therefore, we call this series to be stationary. But if this condition is not satisfied then the series is said to be non-stationary. If $|\beta_{1}|=1$, then the series is non-stationary and we say that there exists a unit root. Our goal in this section is to know whether this beta is significantly different from unity or not. This was a simple example of an AR(1) model, a real-life time series model may be more complex than this. In our testing of unit root, we do not assume that our series follows AR(1) model, the discussion here is for exposition purposes only.
We use the Augmented Dickey-Fuller (ADF) test to test for unit roots in both of our series. The null hypothesis of the ADF test is :

Therefore, if we find p-value to be greater than 0.05, then we cannot reject the null hypothesis at a 95% level of significance. It will mean that our series has a unit root and hence is not stationary.

The result shows that the p-value is greater than 0.05, therefore, we cannot reject the null hypothesis that both series are non-stationary. Therefore, both of our series are not stationary. Hence, the correlation that we saw in EDA section cannot be relied upon.

We generate a first-differenced time series in order to do away with unit root in the original series. The first-differenced series ($\Delta y_{t}$) looks is generated by taking a difference of one period lag from the current period value. That is:

Because the first-differenced time series subtracts the one-period lag, there is no $\Delta y_{1}$ exists because we don’t have $y_{0}$. Therefore, one missing value will be introduced when we take the first difference. Therefore, we need to delete one observation in order to do further analysis. After doing so, we test whether the transformed series by first-differencing is stationary or not.
We again perform ADF test on the transformed series to check their stationarity. The results are reported below (Table-5 ):

The interpretation of the results is that we note that both of our first-differenced series $\Delta y_{t}$ and $\Delta x_{t}$ are stationary at 99% confidence level. In other words, with more than 99% confidence, we can say that the first-differenced series $\Delta y_{t}$ is stationary. The same line applies for $\Delta x_{t}$. Now since our series are stationary, we can apply the usual time series models.

Knowledge of the modeling structure of a series always helps us in our analysis. It is like approaching something with some information about it versus just hitting the target in the dark. Therefore, we go ahead and determine the order of auto-regression and moving average in our transformed time series.
To determine the order of the transformed series, we use the in-built function in $R$. The function that directly gives the ARIMA order a series is auto.arima(), one can see the detailed documentation of the method on the official website of R by clicking here. We employ auto.arima function to get the AR and MA orders of the transformed and original series. We tabulate the results in Table-6

In the table-6, AR-order means up to how many lag terms of itself a series depends on. The I-order means the integration order i.e. it counts a number of unit roots in the time series. If the number or unit roots or I-order is zero, then our series is said to be stationary. The MA order tells us about the structure of the error term. An MA-order being equal to zero means the error term is white noise.
We note that the I-order of the original $y_{t}$ and $x_{t}$ series is one for the transformed series, it is zero. Therefore, we can confirm our earlier findings that both the original series have one unit root each. Confirmation of our findings through an alternate method gives more confidence in our results.

We start this section by discussing why stationarity is important and its definition. We then describe how to test for stationarity. Implementing the stationarity test, we learned that both of our series i.e. Tech index and Non-tech index are non-stationary. In the following text, we transform our series to make it stationary. We note that both series become stationary after taking the first difference. We then confirm this finding by an alternate in-built function in R.

Finally, we close this section by noting that the Tech index ($y_{t}$) follows ARIMA(3,1,3) model, and the Non-tech index ($x_{t}$) follows ARIMA(2,1,4) model.

By testing coupling we mean whether the two series are close enough to each other. We call them decoupled if we can find statistically significant differences between their level in a given period of time. Here we employ a basic t-test to check whether the mean of the two indices is significantly different from each other. Suppose the mean of the tech index is $\mu_{y}$ and the mean of the non-tech index is $\mu_{x}$. Then our null hypothesis is $\mu_{y}=\mu_{x}$. To make it a simple t-test, we take the difference of the two series and test the mean equal to zero. That is our hypothesis is:

We test the null hypothesis of the difference of the two series equal to zero against the alternate of it being non-zero. If we can reject the null hypothesis then we will able to say that the two series are not coupled in levels.

In this section, we split the dataset into different parts. We split our data into four parts covering different data points. The first period covers observations from March 2006 to Feb 2010, the second division covers March 2010 to Feb 2015, the third division covers April 2015 to March 12, 2020, and the last part covers the time after January 20, 2020, to March 2023. We choose a specific date for partitioning the data into third and fourth parts because, as per Center for Disease Center website, the first case of covid in the United States came on this date. We choose this date for splitting the data into the third and fourth parts. We use March 12, 2020 date to do a sensitivity analysis. March 12, 2020, is the date when the first ever stay-at-home order came into existence in the United States. After splitting the data into four parts, we do a t-test to compare the mean of volatility in two series. We further divide the fourth division into two parts: We call period from Jan 20, 2020 to March 2022 to be covid-time. After March 2022, the third wave in the US got over [CDC ref], so we call the period after this time to be post-covid.

We use the stationary transformation of tech and non-tech index and then take their difference to test for coupling. The following table-8 reports the results of hypothesis testing for multiple time periods.

The inferences from these results can be written in the following bullet points:

• •

  The tech and non-tech indices are coupled during 2006-2015. The global financial crisis couldn’t break this coupling.

• •

  The pre-covid period (2015-2020) decoupled the tech and non-tech sectors of the economy. The index levels of the two sectors are significantly different from each other at 95% confidence level. In particular, at 95% confidence, we can say that the tech index level is higher than the non-tech index during the pre-covid period.

• •

  Contrary to popular belief, the covid-19 period does not contribute to decoupling.

Volatility is a measure of ups and downs in a time series. In other words, the spread of a series could be a good measure of ups and down. One such measure is the standard deviation. We use the rolling window method to determine a series’s standard deviation and call it a volatility measure. This measure is a standard practice in the literature. The following steps describe the construction of volatility:

• •

  Step-1: Choose a window size– The first step is to choose a window size, which determines the number of historical returns used to calculate the volatility measure. We use the size of the window equal to 100. This is 100 trading days, which is approximately 4 months.

• •

  Step-2: Calculate returns– Next, calculate the returns for the asset over the window period. The return is typically calculated as the percentage change in price over the window period.

• •

  Step-3: Calculate standard deviation– Calculate the standard deviation of the returns over the window period. This is the volatility measure for that period.

• •

  Step-4: Shift the window– Move the window forward by one day and repeat steps 2 and 3. This will give you a new volatility measure for the next window period.

• •

  Step-5: Repeat– Continue shifting the window forward and recalculating the volatility measure until you have covered the entire time period of interest.

Our choice of window size is an arbitrary one. Generally, a longer rolling window size will provide a more stable and smoother volatility estimate, but at the cost of potentially missing short-term changes in volatility. On the other hand, a shorter rolling window size will be more responsive to short-term changes in volatility, but at the cost of increased noise and potential instability in the volatility estimate. Since our analysis is long-term (17 years), having a rolling window of 100 trading days which is around four months in a calendar sense is justified. We later do a sensitivity analysis by comparing the volatility measures produced by different window sizes to compare qualitative results.
Since we are using a window size equal to 100, we lose 100 data points because there are no previous 100 values available for initial observations to form a window, therefore no volatility measure is constructed corresponding to those data points. However, note that there is one volatility data point for each of the remaining observations. Hence, the total number of volatility series will be equal to 100 less than the number of observations in the original series. For the rest of the discussion, we call the volatility series of the technology index vol_y and the volatility series of the rest of the economy index vol_x.

In our discussion on stationarity, we learned the importance of having a stationary series. Basically, in order to apply the majority of the time series functions, we need to transform the volatility series constructed into a stationary series. Before jumping into whether the volatility series is stationary or not, we plot both volatility series on the time axis.

From the inspection, the series does not appear to be stationary. To formally confirm this guess, we do a statistical test. In the table-9, we describe the appropriate time series model for the two volatility series.

We note that both the volatility series are non-stationary. They are integrated of order one. We close this section by saying that the volatility of tech index vol_y follows the ARIMA(1,1,2) model and the volatility of non-tech index vol_x follows ARIMA(3,1,3)

In the previous sub-section, we noted that both the volatility series have one unit root each. Therefore, a straightforward step to make it stationary is to take the first difference. We have already done so in the EDA section when we were dealing with series $y_{t}$ and $x_{t}$.

After taking the first difference of the volatility series, we do an Augmented Dickey-Fuller (ADF) test to check whether the transformed series is now stationary. For both the series, at 99% confidence, we can reject the null hypothesis that series are non-stationary. Therefore, the transformed series are stationary.

A simple $t$-test compares the means of two samples. Suppose the mean volatility of series vol_y is $\mu_{vol\_y}$ and the mean volatility of series is vol_y is $\mu_{vol\_y}$. The hypothesis of our t-test is:

An alternate way to test this hypothesis is to just take the difference between the two series and test the mean of the new series equal to zero. That is, our null hypothesis is:

Since the assumption of the t-test requires us to have data be stationary, we used the first-differenced volatility series. It is because the original volatility series is non-stationary which doesn’t satisfy the assumptions of t-test, therefore may yield wrong results. Therefore by vol_y we mean the first differenced vol_y in our testing.

We report the t-statistics and corresponding p-value in the table-11.

We observe that we cannot reject the null hypothesis that the difference of means of two volatility series is different from zero at a 95% confidence level. Therefore we conclude that in the full sample, the volatility of tech and non-tech indices are not significantly different.

After all the pre-processing and making our series stationary, we are now ready to apply the standard functions in time series. Note that without making series stationary, we were not able to use this model. Since all the pre-processing is already done, this section is going to be very short one for us. We introduce the model and then discuss the inferences.
Our task is to test whether the Tech index series $y_{t}$ leads the Non-tech index $x_{t}$ series. Suppose that the series $y_{t}$ leads the $x_{t}$ by $k$ time periods. Following time series models of leading indicators as described in Harvey(2020), we basically need to estimate the following equation:

Here, our main coefficient of interest is the vector of $\beta$s. If this vector $\beta$ is significantly different from zero then we can say that the technology index leads the non-technology index. Therefore, we can formulate the null hypothesis as:

Note that we cannot directly use $y_{t}$ or $x_{t}$ in model because these series are not stationary and therefore does not satisfy the assumption needed to use the model. Therefore we use the first differenced series $\Delta y_{t}$ and $\Delta x_{t}$ because they are stationary series.

For simplicity, we used the number of lag to be $p=10$, however we checked for multiple value of $p$, the qualitative results remain the same. The F-statistics of this hypothesis test is $F(10,4252)=1096$, which gives a p-value much lower than 0.001. This means that we can conclude that there is good enough evidence to conclude that the Tech index (series $y_{t}$) can be a leading indicator for the Non-tech index (series $x_{t}$).

One should be careful in interpreting our results, we are saying that $y_{t}$ can be an indicator to lead $x_{t}$, that is what will happen in $x_{t}$ in the future, we can tell by what is happening in $y_{t}$ today (and some previous periods). However, this relationship cannot be claimed as causal. As they say correlation does not imply causation, one may not interpret these results as $y_{t}$ causing the future $x_{t}$. Being a leading indicator is a much weaker condition, it can be established without causation. We also checked whether a reverse relation exists i.e. whether $x_{t}$ can be a leading variable for $y_{t}$, and we found that there is indeed a reverse relationship. Therefore, establishing causality will be a very difficult challenge in this context.

One can raise the question that whether our analysis is valid for the entire economy. This question basically arises from the fact that we are considering a handful number of companies included in the NASDAQ100 index. There are only 100 total companies in this index and out of these 100, both the technology sector index ad the non-technology sector index is constructed. The point is that there may be more companies out there in the technology sector which are not included in NASDAQ100. Similarly, there could be other non-technology companies outside NASDAQ100. What is happening to those companies doesn’t appear in our tech and non-tech indices. Therefore, in some sense what we conclude here in this paper may not be true for the entire technology sector. It could just be true for NASDAQ100 companies which at best are a subset of the entire economy.

We have covered a seventeen-year period which essentially covers the entire lifespan of the NASDAQ100’s technology index sector. Therefore from a time period point of view, our results essentially cover the entire population.

An influential paper on the economics of firms Gabaix (2011) says that firm-size distribution follows something known as power-law distribution. This distribution says that there are very few companies with very high valuations and a very large number of companies with low valuations. The consequence of this is that broader variation in the economy can be explained by the variation in big companies. Gabaix (2011) shows that “The idiosyncratic movements of the largest 100 firms in the United States appear to explain about one-third of variations in output growth”. Since NASDAQ100 contains top-100 companies, therefore, we can say that our findings can well be valid for the entire economy. The basic logic is a large portion of the value of a particular sector is contributed by top companies. An alternate response could be to include more companies. One can consider Russell2000¹¹1The Russell 2000 Index is a stock market index that tracks the performance of approximately 2,000 small-cap companies in the United States. companies to see if there is any difference at the small-cap level. However, including more companies brings in idiosyncratic noise as well, therefore the inclusion of more companies may not always be a good idea.

Given Gabaix(2011)’s result, we stick to NASDAQ100 index and have confidence that our results are valid for the entire economy.