In this article, we will solve and deeply explain two challenging interview questions. You’ll learn how to interpret autocorrelation spikes in time series data and how to critically assess the relationship between correlated variables, such as advertising spend and sales. The explanations will cover underlying concepts, practical steps, and advanced modeling approaches.

Quant Interview Questions from QRT – Solved and Explained

1. Interpreting Autocorrelation Spikes in Daily Time Series

Question:

You calculate the autocorrelation of a daily time series and observe large spikes at lags 7, 14, 21, and 28. What might this tell you about the underlying process, and what would you investigate next?

Understanding Autocorrelation in Time Series

Autocorrelation measures how a time series relates to its past values at different lags. The autocorrelation function (ACF) at lag k is:

$$ \rho_k = \frac{E[(X_t - \mu)(X_{t-k} - \mu)]}{\sigma^2} $$

If a time series is purely random (white noise), we expect autocorrelations to be near zero at all non-zero lags. However, structured processes often exhibit significant autocorrelations at specific lags.

Interpreting Spikes at Lags 7, 14, 21, 28

Observing significant autocorrelation at these lags in daily data is a textbook indication of weekly seasonality:

• Lag 7: One week
• Lag 14: Two weeks
• Lag 21: Three weeks
• Lag 28: Four weeks

This pattern suggests that the time series repeats itself, or has similar behavior, every 7 days. Common examples include retail sales (weekly shopping patterns), website visits (weekday/weekend effects), or energy usage (workweek cycles).

Key Follow-Up Investigations

• 1. Check for Day-of-Week Effects:
  Does the process behave differently on different days? For example, are Mondays consistently higher than Sundays? You can visualize this by grouping data by day of week:

  
  import pandas as pd
  import matplotlib.pyplot as plt
  
  df['day_of_week'] = df['date'].dt.day_name()
  df.groupby('day_of_week')['value'].mean().plot(kind='bar')
  plt.show()
      

• 2. Seasonal Differencing:
  If the series displays strong seasonality, seasonal differencing can help make it stationary. This involves subtracting the value from one week ago:

  
  df['diff_7'] = df['value'] - df['value'].shift(7)
      

  After seasonal differencing, recalculate the ACF to check if the seasonality is removed.
• 3. Fourier Terms:
  Periodic effects can be modeled using Fourier (sine/cosine) terms:

  
  import numpy as np
  df['sin_7'] = np.sin(2 * np.pi * df.index / 7)
  df['cos_7'] = np.cos(2 * np.pi * df.index / 7)
      

  These can be used as regressors in linear or time series models.
• 4. SARIMA or Seasonal Models:
  Seasonal ARIMA (SARIMA) models explicitly model periodicity. The SARIMA(p,d,q)(P,D,Q)[s] model, where s is the seasonality period (7 for weekly), is often appropriate. The SARIMA equation:

  $$ \Phi_P(L^s) \phi_p(L)(1-L)^d(1-L^s)^D y_t = \Theta_Q(L^s) \theta_q(L) \epsilon_t $$

  Where:
  • \( \phi_p(L) \): non-seasonal AR
  • \( \theta_q(L) \): non-seasonal MA
  • \( \Phi_P(L^s) \): seasonal AR
  • \( \Theta_Q(L^s) \): seasonal MA
  • \( (1-L)^d \): non-seasonal differencing
  • \( (1-L^s)^D \): seasonal differencing

  
  from statsmodels.tsa.statespace.sarimax import SARIMAX
  model = SARIMAX(df['value'], order=(p,d,q), seasonal_order=(P,D,Q,7))
  result = model.fit()
      

• 5. Visualize ACF and PACF:
  Use autocorrelation and partial autocorrelation plots to confirm the presence and length of seasonality:

  
  from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
  plot_acf(df['value'], lags=30)
  plt.show()
      

Summary Table: Steps to Investigate Seasonality

Advanced Considerations

• Check for Calendar Effects: Beyond weekly cycles, check for monthly, quarterly, or holiday effects.
• Multiple Seasonality: Some data may have both weekly and yearly cycles (e.g., daily retail sales), which may require specialized models (e.g., TBATS or Prophet).

Conclusion

Spikes in autocorrelation at multiples of seven in daily data are strong evidence of weekly seasonality. Robust time series modeling involves not just detecting, but also quantifying and adjusting for these effects using differencing, Fourier terms, or explicit seasonal models like SARIMA.

2. Interpreting High Correlation Between Advertising Spend and Sales

Question:

You have two daily time series: advertising spend and sales. Their correlation is 0.85. Your manager concludes that advertising is strongly driving sales. Would you agree with that conclusion? What would you investigate before making that claim?

Concepts: Correlation, Causality, and Time Series Pitfalls

A correlation of 0.85 between advertising spend and sales seems impressive, but correlation does not imply causation. Especially in time series data, naïvely interpreting correlation can lead to misleading or even spurious conclusions.

Problems with Correlation in Time Series

• Non-Stationarity: If both series have trends or seasonal patterns, they can appear highly correlated even if there is no causal link. This is known as “spurious correlation.”
• No Directionality: Correlation does not tell us whether advertising drives sales, or vice versa.
• Confounding Variables: Both advertising and sales could be driven by a third factor (e.g., holidays, economic conditions).

Key Investigations Before Drawing Conclusions

1. 1. Check for Trends and Seasonality in Each Series
   • Plot each series to visually inspect for upward or downward trends and repeating seasonal patterns (e.g., weekends, holidays).
   • Use statistical tests for stationarity, such as the Augmented Dickey-Fuller (ADF) test.

   • 
     from statsmodels.tsa.stattools import adfuller
     adf_ad = adfuller(df['advertising'])
     adf_sales = adfuller(df['sales'])
     print("Advertising ADF p-value:", adf_ad[1])
     print("Sales ADF p-value:", adf_sales[1])
             

   • If either series is non-stationary, their correlation is likely inflated.
2. 2. Difference the Series to Remove Trends/Seasonality
   • If both series are integrated (non-stationary), difference them (subtracting previous day’s value) to make stationary:

     
     df['ad_diff'] = df['advertising'].diff()
     df['sales_diff'] = df['sales'].diff()
             

   • Re-calculate the correlation on the differenced data. If the correlation drops sharply, the original high correlation was likely spurious.
3. 3. Check for Lagged Effects (Does Advertising Lead Sales?)
   • Advertising may affect sales after a delay. Compute cross-correlation at various lags:

   • 
     from statsmodels.tsa.stattools import ccf
     import numpy as np
     lags = np.arange(-14, 15)
     ccf_values = [df['advertising'].shift(lag).corr(df['sales']) for lag in lags]
     plt.plot(lags, ccf_values)
     plt.xlabel('Lag (days)')
     plt.ylabel('Cross-correlation')
     plt.show()
             

   • If advertising at lag k days correlates with sales today, this supports a causal interpretation.
4. 4. Control for Confounding Variables
   • Other factors (e.g., promotions, holidays, general market trends) may drive both ad spend and sales.
   • Include these as control variables in a regression or time series model.
5. 5. Granger Causality Test
   • Granger causality tests whether past values of advertising help predict sales, beyond what past sales alone can predict.

   • 
     from statsmodels.tsa.stattools import grangercausalitytests
     grangercausalitytests(df[['sales', 'advertising']], maxlag=7)
             

   • A significant p-value for advertising causing sales at some lag supports the causal claim.
6. 6. Regression with Lags and Controls
   • Fit a regression model with lagged advertising spend and other confounders as predictors.

   • 
     import statsmodels.api as sm
     df['ad_lag1'] = df['advertising'].shift(1)
     df['ad_lag2'] = df['advertising'].shift(2)
     X = df[['ad_lag1', 'ad_lag2', 'other_controls']]
     y = df['sales']
     model = sm.OLS(y, sm.add_constant(X), missing='drop').fit()
     print(model.summary())
             

   • Significant coefficients on lagged ad spend, after controlling for other variables, strengthens the case for causality.

Example: Spurious Correlation Due to Trend

Suppose both advertising and sales increase over time due to company growth. The raw correlation will be high, even if advertising has no effect on sales. Differencing removes the trend, revealing if there’s a true relationship.

Summary Table: Steps for Causal Analysis

Advanced: Cointegration

If both series are non-stationary but move together over the long term, they may be cointegrated. Cointegration analysis and error correction models can be usedto model the long-run equilibrium relationship and short-run dynamics between the two series. Cointegration is a critical concept in time series econometrics, especially when analyzing economic or financial data that may drift but remain linked by an underlying equilibrium.

What is Cointegration?

Suppose two non-stationary time series, \( X_t \) (advertising spend) and \( Y_t \) (sales), both exhibit trends over time. Even if each series by itself is non-stationary, their linear combination may be stationary:

$$ Z_t = Y_t - \beta X_t $$

If \( Z_t \) is stationary, the series are said to be cointegrated. This suggests a meaningful long-term equilibrium relationship between sales and advertising, even if short-term deviations exist.

How to Test for Cointegration

• Engle-Granger Two-Step Method:
  1. Regress sales on advertising: \( Y_t = \alpha + \beta X_t + \epsilon_t \)
  2. Test the residuals (\( \epsilon_t \)) for stationarity using the ADF test.

  
  import statsmodels.api as sm
  reg = sm.OLS(df['sales'], sm.add_constant(df['advertising'])).fit()
  residuals = reg.resid
  from statsmodels.tsa.stattools import adfuller
  print('ADF p-value on residuals:', adfuller(residuals)[1])
      

  • If the residuals are stationary (ADF p-value < 0.05), the series are cointegrated.
• Error Correction Model (ECM):
  • If cointegration is found, model the short-term dynamics and the adjustment towards equilibrium with an ECM:

    $$ \Delta Y_t = \gamma_0 + \gamma_1 \Delta X_t + \gamma_2 (Y_{t-1} - \beta X_{t-1}) + \epsilon_t $$

  • The coefficient \( \gamma_2 \) measures the speed at which sales revert to the long-run relationship with advertising.

Interpretation for Interviews

A strong candidate will recognize that a high correlation between two trending (non-stationary) series may simply reflect their shared trends, not a true causal relationship. By checking for stationarity, differencing, and cointegration, you demonstrate advanced understanding of time series relationships and avoid making unsupported claims about causality.

Putting It All Together: Workflow for Attribution Analysis

1. Plot and visually inspect each time series.
2. Test for stationarity; if non-stationary, difference the series.
3. Recalculate correlation on differenced data.
4. Test for cross-correlation at various lags to determine directionality.
5. Test for Granger causality to see if past advertising predicts sales.
6. Control for confounding variables in multivariate regressions.
7. If both series are non-stationary, test for cointegration and, if present, use an error correction model.

Common Pitfalls and Best Practices

• Avoid Spurious Regression: Never regress non-stationary series on each other without first addressing trends or cointegration.
• Consider Lagged Relationships: Advertising effects may not be immediate; use cross-correlation and lagged regressions.
• Account for Seasonality and Calendar Effects: Both advertising and sales may rise during holidays or weekends, creating spurious associations.
• Include Control Variables: Always consider third variables that might influence both advertising and sales.

Conclusion: Mastering Quantitative Interviews at QRT

The ability to interpret time series patterns and critically assess relationships between variables lies at the core of quantitative research roles. In this article, we covered two advanced interview questions you might encounter at QRT or similar quantitative research teams.

• For autocorrelation spikes at lags 7, 14, 21, 28: Recognize weekly seasonality, investigate using differencing, Fourier terms, and SARIMA models, and always confirm with visual and statistical tools.
• For high correlation between advertising and sales: Understand the dangers of spurious correlation, use stationarity tests, differencing, cross-correlation, Granger causality, regression with controls, and cointegration analysis to build a robust causal argument.

By demonstrating these lines of inquiry, you show not only technical skill but also the critical, skeptical mindset valued in top quantitative research teams. Remember: always probe beyond the surface, question your assumptions, and let the data guide your conclusions.

Further Reading and Resources

• Forecasting: Principles and Practice (Hyndman & Athanasopoulos) – Comprehensive guide to modern time series analysis.
• Statsmodels Documentation – Python library for statistical modeling and time series.
• Time Series Analysis in Python – Practical tutorial with code examples.
• Wikipedia: Granger Causality
• Spurious Regression and Cointegration (Duke University)

FAQ: QRT Quant Researcher Interview Questions

Summary

Excelling in QRT quant researcher interviews is about more than memorizing formulas – it’s about understanding the nuances of time series data, questioning apparent relationships, and applying rigorous statistical reasoning. By mastering the concepts and techniques discussed here, you'll be well prepared for advanced quantitative research interviews and practical modeling challenges in the field.

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