Has the Asymmetric Effect of Oil Price Change in Inflation Expectations Been Impacted by the COVID-19 Outbreak? A Comparison Between the United States and China

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Introduction
Oil is one of the important non-renewable energy sources, and it also plays an irreplaceable role in world production activities, resulting in oil prices being closely related to the development of the world economy.With the continuous spread of the COVID-19 epidemic and the sustained higher oil prices since the end of 2020, the impact of oil prices on inflation is particularly prominent.Given the interaction of inflation and inflation expectations (Lagoa, 2017;Marfatia, 2018), oil price shocks can also significantly affect inflation expectations.Therefore, this paper aims to study the changes in inflation expectations in the United States and China affected by oil price shocks before and after the COVID-19 outbreak, and the changes in the asymmetry of oil price shocks in the short and long term.
Because of the importance of oil, oil prices have always been the focus of scholars.With the advent of more research on the relationship between oil prices and other macroeconomic variables, we find that oil price changes have effects on production cost (Bresnahan & Ramey, 1993), economic growth (Nie, 2023), income (Bohi, 1991), consumption expenditure (Kilian, 2008;Nie, 2023), and unemployment (Davis & Haltiwanger, 2001).In the United States and China, as the largest importers of crude oil, scholars' research on topics related to oil shock has never stopped since the 1970s.Blanchard and Galí (2010) studied the difference in the impact of oil price shocks on inflation and economic activity in the United States in the 1970s and after 2000, arguing that market structure, monetary policy, oil's share in the market, and uncertainty policies contribute to this difference.Coibion and Gorodnichenko (2015) used the rise in oil prices to explain the absence of deflation in the United States during the Great Recession, and the rise in inflation expectations was also one of the transmission mechanisms of the effect of oil prices on inflation.Chen et al. ( 2020) also discovered the transmission mechanism of decomposed oil shocks to Chinese inflation.Li and Guo (2022) used the Nonlinear ARDL model with multiple thresholds to find that oil price shock has a significant asymmetric effect on Chinese inflation in the short term, which indicates that the inflation effect is more significant when oil price falls.Inflation and its important driver, inflation expectations (Armantier et al., 2020;Istiak & Alam, 2019a), are both affected by oil price changes (Armantier et al., 2016).At present, economists and scholars generally recognize the existence of oil price shocks, which are affecting inflation mainly by five mechanisms: discretionary income effect, uncertainty effect, precautionary savings effect, operating cost effect and reallocation effects (Edelstein & Kilian, 2009;Pal & Mitra, 2019).The most direct impact of oil price shocks on Chinese and American households' inflation expectations is the change of disposable income.
In recent decades, there has been extensive literature exploring the importance and determinants of inflation expectations.The Reserve Bank's Second Quarter Review of Monetary Policy 2012-13 noted that a central premise of monetary policy is that low and stable inflation and well-anchored inflation expectations contribute to a conducive investment climate and consumer confidence, which is key to sustained growth on a higher trajectory in the medium-term.The survey-based approach to measuring inflation expectation presents both short-term and long-term forecasts on inflation and covers different target groups, including households or professional forecasters.Pfajfarabc and Žakeljd (2014) think that Central banks increasingly attribute more importance to the developments of households‫׳‬ inflation expectations as they signal future inflationary risks.Mohanty (2012) proposed that there are two ways of forming inflation expectations.The first is a variant of adaptive behavior wherein expectations are formed by extrapolating past and current experiences into the future.The second way of forming inflation expectations in a forward-looking manner is rational expectations.D 'Acunto et al. (2021) propose that the changes in prices affect inflation expectations, with positive changes in prices having a greater effect than negative changes.Liu (2019) agrees that economic policy uncertainty works on China's inflation expectations as well.Ueda (2010) also includes that exogenous prices and monetary policy shocks have significant effects on the formation of household inflation expectations by exploring household inflation expectations in the United States and Japan.Zhang (2011) explores the impact of monetary policy on Chinese household inflation expectations.Ghosh et al. (2021) point out that India's Inflation expectations are determined by factors such as output, inflation rate, monetary policy, exchange rate, economic policy uncertainty, oil prices, and financial volatility.Kortelainen et al. (2011) apply the conventional New Keynesian macro model to the estimation of inflation expectations in Europe and the United States, and conclude that the use of measurement expectations greatly reduces the impact of lagged output and inflation terms on economic forecasts' importance.Aristidou (2018) analyzes the US data from 1950-2016 through the Phillips curve, arguing that although the importance of future inflation expectations will change over time, depending on the monetary policy regime and economic environment, future expectations contribution to current inflation is greater than past inflation.
By affecting the real interest rate, inflation expectations have an impact on the intertemporal decisions of private households and firms on savings, consumption, and investment.There are two central functions that inflation expectations can perform in central banking.First, as important inputs into price and wage setting, they provide a summary statistic of where inflation is likely to be headed.Second, they may be used to assess the credibility of the central bank's inflation objective.More scholars are realizing the importance of inflation expectations to economic development.By analyzing inflation expectations in the United States and Europe, Ciccarelli, and Garcia (2015) find that inflation expectations have spillover effects across countries, which suggests a need for coordinated policy actions, mainly in times of crisis; Lagarde (2020) suggests the ability to manage households' inflation expectations is key for the effectiveness of monetary and fiscal policies.The impact of oil price shocks on inflation expectations has also attracted more attention from scholars.Cao and Shapiro (2016) point out that oil prices have pervasive effects on inflation expectations with breakeven inflation or inflation swap rates.Hammoudeh and Reboredo (2018) propose that U.S. oil prices have non-linear effects on market-based 5-and 10-year inflation expectations components, which are more prevalent in the medium term than in the long run.Kilian and Zhou (2021) predict that rising oil prices in 2021 will have an impact on US household inflation expectations, with a peak response of 1.2 percentage points for 1-year household inflation expectations and a peak response of 0.2 percentage points for 5-year expectations.
Affected by the epidemic, the uncertainty of economic development in the United States and China has increased.Two of the most notable results are the rising of households' inflation expectations and oil price shocks (Apergis, 2021;Tan et al., 2022).All the study, related to the impact of oil price shocks in the United States and China on inflation expectations and the effect of the epidemic on the relationship between oil price changes and inflation expectations, is conducive to helping inflation expectations play a better anchoring role and explaining the differences in economic development before and after the epidemic.To reflect reasonable short-and long-term asymmetries in oil price dynamics at the same time, we apply ARDL and Nonlinear ARDL frameworks to two long-term inflation-targeting policy mechanisms (Nasir, 2020), using the Phillips curve to assess the relationship between oil price changes and inflation expectations (Eser et al., 2020) and the asymmetric differences before and after the epidemic.
This paper mainly has the following four contributions.We use a reasonable method to convert quarterly qualitative data on China's residents' inflation expectations into quantitative data and substitute monthly inflation expectations with reasonable indexes based on previous research and finally study them as dependent variables, which provides the basis and conditions for future research on Chinese residents' inflation expectations.Based on existing research, we use the Carson-Parkin Method to convert China's only data reflecting residents' inflation expectations from qualitative data into quantitative data.For further robustness checks, we refer to and use the arithmetic average of real interest rates and lagged inflation rates to show residents' monthly inflation expectations.Please refer to the data description section for specific instructions.Second, this is a complement to the ARDL and NARDL models under different regimes.Most previous studies only considered the impact of shocks within a specific time period, and rarely explored the comparison under different regimes.Some previous scholars have used the NARDL model to compare the different effects of oil shock under pre-ZLB and ZLB regimes on the inflation rate in New Zealand.However, our paper is the first to focus on the different effects of oil price changes on inflation expectations under different regimes, pre-COVID and COVID periods.Third, to the best of our knowledge, this article is the first to compare the differences in the asymmetry of oil price shocks between China and the United States before and after the epidemic.As the U.S. and China are the world's largest importers of crude oil and the countries most affected by the COVID-19 outbreak, it is of great significance to research this topic but there is limited research that explores the role of the COVID-19 pandemic event in the impact of oil prices on inflation expectations.We solved the problem of fewer samples during the epidemic by introducing a dummy variable and interactive variable, so as to see the change in the effect, and conducted a robustness test through high-frequency data to achieve a comparison of the asymmetric impacts in the U.S. and China before and after the epidemic.
The structure of this article is as follows.In Section 2, we explain the data and methodology of the study.In Section 3, we provide preliminary test results for model suitability.A comparison of empirical results between the United States and China is presented in Section 4. In Section 5, we examine the causality of the variables and the robustness of the results.Section 6 further discusses and concludes the results.

Data
As mentioned in the introduction, to explore the impact of oil prices on inflation expectations, this paper uses data from the United States and China.The reasons for choosing these two countries are as follows: First, China and the United States are currently the importers with the largest volume of crude oil in the world.The impact of crude oil price shocks on their inflation expectations is more significant, exploring which is of great significance for the formulation of policies in the two countries and the future consumption structure of consumers; second, as countries with the largest GDP in the world, both the United States and China economies have suffered from the COVID19 epidemic.By comparing the impact of oil price shocks under the two regimes of pre-COVID19 and COVID19, we can greatly reflect the world trend of changes in the impact of oil price shocks before and after the epidemic.Third, the United States is in a free market economic system, while China adopts an economic system in which open competition and government regulation coexist.Research on the impact of oil price shocks before and after the epidemic also reflects the impacts of different economic systems under special circumstances, which has significance in studying the effectiveness of different economic systems and policies.
To analyze the impact of potential determinants, including oil prices, on inflation expectations, we draw on the approach of Nasir et al. (2020) to designate lagged inflation expectations, deflated crude oil prices, actual inflation rate, seasonally adjusted GDP, and economic policy uncertainty.Given that these factors are under standard theories and are often empirically identified as the main determinants of inflation, we emphasize that these factors are influential factors in inflation expectations and their dynamics.It is worth noting that according to the analysis from Jose and Frederick (2006), they support the presence of a cointegrating relationship between the crude oil and natural gas price time series, providing significant statistical evidence that WTI crude oil and Henry Hub natural gas prices have a long-run cointegrating relationship.Natural gas is also an important source of energy for residential heating.Crude oil is one of the most important components of gasoline, and the price of gasoline is influenced by crude oil, which is closely related to people's lives, and is one of the criteria by which people measure price levels and change inflation expectations.Given the importance of crude oil, natural gas, and gasoline in households' daily lives, we will use the same Nonlinear ARDL model but replace the endogenous variables crude oil price and gasoline price as follows natural gas price and gasoline price, respectively, in order to investigate the existence of asymmetry in the impact of price changes in gasoline and natural gas on inflation expectations.
Considering the continuity and availability of data, we use quarterly and monthly observations from January 2010 to December 2021 for data analysis.First, we will use quarterly observations to draw overall conclusions, but considering the limited post-pandemic data, we will take monthly data for robustness check.The details of the variables are as follows:

Inflation expectations:
There are two main sources of data on inflation expectations, market-based inflation expectations, and survey-based inflation expectations.This article focuses on the latter, and more specifically, the survey data on inflation expectations with households as respondents.
The U.S. inflation expectations data is based on households' inflation expectations provided by the Michigan survey.The inflation expectations of consumers from the University of Michigan's Survey of Consumer Attitudes and Behavior.In this survey, the data on consumer sentiment are collected by interviewing a random sample of approximately 500 U.S. households each month.The consumer sentiment represents the forecast of the respondents About some key macroeconomic variables, such as inflation, interest rates, and unemployment.We get the monthly and quarterly data of inflation expectations from the Federal Reserve Economic Database maintained by the Federal Reserve Bank of St Louis.All the data is quantitative.
China's quarterly inflation expectations data are based on the quarterly survey given by the People's Bank of China for more than 20,000 savings users in 50 different cities across the country since 1999 (the Urban Depositor Questionnaire Report of the Statistics and Analysis Department of People's Bank of China1 ).With three options, i,e, up, unchanged, and down, the survey gives us residents' qualitative views on the change in CPI over the next three months, as well as the percentage of each option.The method in this paper to convert these qualitative data into quantitative indicators is C-P Method (Carlson & Parkin, 1975).The basic principle of this method is: Assuming that respondents' expectations for future price level changes are subject to a specific probability distribution, and that there is a "sensibility interval" centered at 0. If the respondent's judgment on the price increase in the next period exceeds the range, "up" is selected, if it falls below the range, "down" is selected, "unchanged" otherwise.Respondents' answers were symmetrical and normally distributed, and the average realized in the past was equal to the expected average.In order to test the convincingness of the inflation expectation calculated by this method, we refer to the correlation test of historical data by many scholars and find that the correlation between the inflation expectation calculated by the C-P method and the price expectation index since 2000 is even as high as 0.78.Chinese monthly inflation expectation is the arithmetic average of the real interest rate and lagged inflation rate, which adopts the method of Yu et al. (2018).Given that there is no direct monthly household inflation expectation in China, many scholars try to reflect residents' inflation expectations through substitutable variables, so as to reflect the impact of residents' inflation expectations on economic development.The Chinese government even proposed for the first time in 2009 to control inflation expectations as one of the focuses of macroeconomic regulation2 .Yu et al. (2018) used a VAR expectation model with additional forward-looking policy variables and a Kalman filter recursive algorithm to confirm that the inflation expectation since 2002 calculated by this method is unbiased, and the mean value of the expected error is zero and there is no autocorrelation.Therefore, this paper also uses this method to calculate monthly residents' inflation expectations, thereby making the possibility for robustness testing.More details in methods references are in Appendix C3 .

Crude oil price:
Crude oil prices in the US and China are the oil prices in real terms deflated RAC by U.S. and China CPI (Elder and Serletis, 2010), in which RAC is the composite refiners' acquisition cost (RAC) of crude oil from the website of the Federal Reserve Bank of St. Louis.The crude oil (petroleum) price is utilized as the proxy indicator of global crude oil price, and the unit is US dollars per barrel.The specific calculation formula is as follows, ).

Inflation rate:
China's monthly inflation rate is calculated from the growth rate of month-on-year CPI data released by the National Bureau of Statistics, and the quarterly inflation rate is obtained after the monthly average, and seasonally adjusted.Inflation in the United States is seasonally adjusted quarterly and monthly year-over-year growth in the CPI index.We get the monthly and quarterly CPI data of CPI from the Federal Reserve Economic Database maintained by the Federal Reserve Bank of St Louis.

GDP:
GDP in this paper is the logarithm of GDP per capita (unit is US billion dollar).U.S. quarterly and monthly GDP data the real GDP from the Federal Reserve Bank of St. Louis from 2010 to 2021, which has been seasonally adjusted.China's quarterly GDP is seasonally adjusted 2010-2021 data from the National Bureau of Statistics of China.Since China does not have direct monthly GDP data, in order to reflect the development trend of monthly GDP as much as possible, this paper substitutes it with the monthly industrial-added value data from the National Bureau of Statistics of China.The growth of China's GDP is mainly due to the growth of the added value of the three industries.The added value of the secondary industry accounts for an average of 41.93% of the added value of the GDP from 2010 to 2021, and the added value of the industry is an important part of the secondary industry.Moreover, according to China's historical data, after 2000, the correlation coefficient between the growth rate of China's quarterly industrial added value and the quarterly GDP growth rate was as high as 0.9264.Therefore, the industrial-added value is an important surrogate indicator for measuring Chinese GDP.

EPU:
Uncertainty of economic policy is an important variable that reflects the economic risk caused by the uncertainty of government policy in the future.Baker et al. (2016) define the EPU index into three main components: the first component quantifies newspaper coverage of policy-related economic uncertainty in major national newspapers by constructing a normalized index of the number of news articles discussing EPU; the second component reflects the number of federal tax code provisions that will expire in the next 10 years to measure the degree of uncertainty about the path the federal tax code will take in the future; the last component uses divergence among economic forecasters as a proxy for uncertainty, Specifically, the differences between individual forecasters' forecasts of the consumer price index, federal spending, and future levels of state and local spending are used to construct uncertainty indices about important macroeconomic variables.U.S. economic policy uncertainty (EPU) index is the monthly and average quarterly data from the Federal Reserve Bank of St.
Louis.China's monthly EPU index is from https://www.policyuncertainty.com/china_monthly.html,developed from Steven J. Davis, Dingqian Liu, and Xuguang S. Sheng's working paper "Economic Policy Uncertainty in China since 1949: Perspectives from Mainland Newspapers".They began quantifying concepts related to uncertainty using two mainland Chinese newspapers, People's Daily and Guangming Daily, starting in October 19494 .China's quarterly data is an average of the monthly EPU index.

Econometric Specification
The main purpose of this study is to investigate the impact of the COVID-19 outbreak on oil price shocks and the impact of oil price shocks on inflation expectations in the United States and China.In the basic model, this paper uses the quarterly data of inflation expectations, oil price shock, inflation rate, GDP, and EPU index from 2010 to 2021 for linear and nonlinear regression analysis.However, considering the limited number of quarterly data, we introduce a dummy variable to show the effect of structural break, and an interactive variable to reflect the difference in oil price shocks before and after the break, so as to ensure the integrity of the sample and the differences on oil dynamics.In a further robustness check, we will conduct the same analysis using monthly data from the United States and China, but the difference is that we divide the sample into the full sample, pre-COVID19 and COVID19 regimes (i.e.full sample with two sub-samples), while excluding the effects of dummy variable and interactive variable on the model, in each sample to analyze the linear and nonlinear effects of lagged inflation expectations, oil price shock, inflation rate, GDP and EPU index on inflation expectations.
First, in order to reflect the relationship between the target endogenous and exogenous variables and households' inflation expectations in the United States and China and, we propose a model as follows: where   stands for inflation expectation(unit is %),   shows oil prices, which is the deflated global crude oil price(unit is U.S. Dollars per Barrel),   indicates real inflation rate(unit is %),   is the logarithm of GDP per capita (unit is US dollar), and   shows economic policy uncertainty.  is the dummy variable used to separate the full sample from 2010Q1 -2021Q4 into two regimes of pre-COVID19 and COVID19, where   = 0 if data was collected in pre-COVID19 regime and   = 1 if data was collected in COVID19 regime.( * )  is the interactive variable, which indicates the volatility of oil prices after the epidemic. 0 is a constant term, representing the lowest inflation expectations in the United States and China. 1 ,  2 ,  3 ,  4,  5 ,  6 ,  7 are the coefficients of variables, reflecting the impact of each variable on inflation expectations. is the symbol of countries, i,e, when  = 1, the country is U.S,; when  = 2, the country is China.

If the coefficient of the dummy variable 𝛿 6𝑓
̂ is not equal to 0 and is meaningful, it indicates that the outbreak of the COVID19 epidemic is an important breakpoint, and the impact of oil price changes on inflation expectations is significantly different before and after this structural break.If the coefficient of the interactive variable  7 ̂ is greater than 0 and meaningful, it means the impact of oil price changes on inflation expectation in the COVID19 regime is bigger than the effect in the pre-COVID19 regime, while if  7 ̂< 0 and have significant, it shows that the impact of oil price changes on inflation expectation in COVID19 regime is smaller than the effect in the pre-COVID19 regime.
Notably, Eq.1 can only deduce the long-term effects of exogenous variables.In order to explore both the short-term and long-term effects of oil price shocks before and after the epidemic, this paper adopts both the autoregressive distributed lag (ARDL) model developed by Pesaran, Shin, and Smith (2001), and the nonlinear autoregressive distributed lag (Nonlinear ARDL) model developed by Shin, Yu, and Greenwood-nimmo (2014).

ARDL-ECM
An autoregressive distributed lag (ARDL) model is an ordinary least square (OLS) based model applicable for both non-stationary time series and times series with mixed order of integration.There are many advantages of the ARDL Model: 1.The model shows the effects from both explanatory variables and lagged dependent variables; 2. by confirming the co-integration relationship between variables, the model can be used to analyze short-term and long-term dynamics; 3. the error estimation of the model is suitable for both small and large samples; 4. the model has relatively less requirements on independent variables, that is, variables can be stationary at I(0) or I(1), or the mixed of both; 5. ARDL model is the only approach that provides us with some explicit tests through which we can explore that exclusive cointegration exists or not instead of assuming vector existence (Nkoro and Uko, 2016).Because of the limited number of samples during the pandemic so far, ARDL Error Correction Model is more robust and suitable for this research.
According to the cointegration theory of Engle (1982); Engle and Granger (1987), if all the variables are I(1) and cointegrated, we can use an error correction model (ECM) to represent the dynamic relationship between variables.In the ARDL model, Pesaran and Shin (1999) proposed that if variables are stationary at either I(0) or I(1) and are cointegrated with each other, we can use ARDL-ECM model to express the dynamic relationship.Especially, ARDL model determines the cointegration relationship more efficiently when the sample size is small.Therefore, if inflation expectations, deflated oil price, inflation rate, GDP, EPU, dummy variable and interactive variable are stationary at I(0) or I(1) and are cointegrated in the long run, we can estimate ECM based on ARDL (ARDL-ECM) as follows: where ( According to the ARDL Error Correction Model, we can explore the short-term and long-term effects of determinants on inflation expectations, analyze whether there is a structural break in the impact of oil price change on inflation expectations before and after the COVID19 outbreak, and if so, what the change on the impact of oil price change is after the break.

Nonlinear ARDL Model
However, fluctuations in global oil prices may have asymmetric and non-linear effects on short-and long-term inflation expectations (Mihajlović & Marjanović, 2020;Long & Liang, 2018), therefore asymmetric and Nonlinear ARDL models are appropriate and necessary for studying the nonlinear and asymmetric relationships.
The nonlinear ARDL model is a single-equation error correction model that can accommodate asymmetry in the long-run equilibrium relationship and/or the short-run dynamic coefficients via the use of partial sum decompositions of the independent variable(s) proposed by Shin, Yu, and Greenwood-nimmo (2014).In the Nonlinear ARDL, since we focus on exploring the asymmetric effect of oil price changes on inflation expectations, we move oil price changes decomposed into its negative and positive partial sums, which makes it possible to test whether oil shock has symmetric or asymmetric effects on inflation expectations in the short-and the long-run.Therefore, the deflated oil price   is expressed as: where  is still the symbol of countries, i,e, when  = 1, the country is U.S,; when  = 2, the country is China.  + and   − are partial sums that capture the increase and decrease of the oil price in U.S and China, expressed as: (5) where ∆  =   − −1 .
After introducing the positive and negative shocks of oil prices into the basic VAR model Eq.1, we rewrite Eq.1 as follows: where  0 is the constant term representing the lowest inflation expectations in the US and China. 1 ,  2 + ,  2 − ,  3 ,  4,  5 ,  6 ,  7 are the coefficients reflecting the effect of variables on inflation expectation in the Nonlinear ARDL model, where  2 + represents the long-term impact of positive oil prices shock on inflation expectations, and,  2 − represents the long-term impact of negative oil prices shock on inflation expectations.By comparing the significant differences between these two coefficients, it can be confirmed that there is an asymmetric impact between positive and negative changes in oil prices in the long term.
Therefore, to introduce both the long-and short-term effects of rising and declining oil price, we apply the Nonlinear ARDL model as follow: All variables are as previously described.k, l, m, n, p, q, r are the optimal lag orders of variables selected by the Akaike information criterion (AIC) and Schwarz information criterion (SIC).To establish the relationship between the positive and negative oil prices shocks and inflation expectations, according to the boundary test proposed by Pesaran, et al., (2001), we believe that the long-term effects of oil price changes are and . The short-term impact of positive oil prices on inflation expectations, ∑  2 +  =1 measures the short-term impact of negative oil prices on inflation expectations.In this case, we capture the asymmetry of oil price changes in the short-run and long-run.
Before that, we also need to confirm that all variables are stationary at I(0) or I(1) or the mixture.It is also necessary to confirm that all variables are not stationary at I(2).This is because I(2) invalidates the calculation of the F statistic to test for cointegration (Ibrahim, 2015).We perform the unit root test with and without structural breaks to find the order of integration.Then, long-run and short-run asymmetry in the nonlinear ARDL model can be tested by the standard Wald test.In the long run, the null hypothesis is  1 =  2 .A significant difference in these two values would affirm an asymmetric relationship in the long run.In the same way, ∑  2 indicates that there is no asymmetry in the short run.Moreover, the existence of cointegration could be assessed using the bounds test where the null hypothesis ( Similarly, we can also implement the error correction model (ECM) of the Nonlinear ARDL model as follows: . 2 + and  2 − indicate the impact of increasing and decreasing oil prices on inflation expectations in the long run. 6 shows the impact difference of oil price shock before and after the epidemic. 7 expresses the impact change on oil price shock after the outbreak.

Descriptive Statistics
The oil price change has an important impact on the economic development of fell from 542 million tons in 2020 to 512.98 million tons, a drop of 5.4%.This also means that China's crude oil imports have declined for the first time in 20 years.In 2020, international oil prices reached the lowest level in decades, and in April 2020, the year-on-year decline in global crude oil demand reached the lowest level since 1995.In 2021, international oil prices continue d to rise, reaching the highest level since 2008.Not only that, but GDP growth has also been seriously affected.In the 10 years from 2010 to 2019, the average annual growth rate of GDP in the United States was 2.25%, and the average economic growth rate in China during the same period was 7.68%.However, affected by the epidemic, the GDP of China and the United States fell by 4.8% and 6.8% respectively in the first quarter of 2020.Simultaneously, the year-on-year inflation rate in the United States increased by 14.1%, and the inflation rate in the second quarter reached 2.1%.In the first quarter of 2020, the year-on-year inflation rate in China was lower than that of the previous quarter, but it was still at 4.36%.The rise in energy and food prices caused by the pandemic has led to rising inflationary pressures in many countries.EPU has also risen significantly.The EPU of the United States in the second quarter of 2020 was as high as 417.3, an increase of 88.1% over the previous quarter.China's EPU growth in the first quarter of 2020 also reached 100.0%.Therefore, the epidemic has greatly exacerbated the uncertainty of the change in oil prices and even the development of the world economy.All economies should study the difference in the impact of the oil shock before and after the epidemic.
To sum up, in the benchmark model, this paper uses 2010-2021 U.S. and China quarterly households' inflation expectations, oil price shocks, inflation, GDP and EPU indices, as well as a dummy variable that represents structural breakpoint and an interactive variable to show the change on the oil prices after the COVID-19 outbreak.Table 1 shows the mean, variance, minimum and maximum values of all variables as follows:

Unit Root Test
As mentioned earlier, to use ARDL and Nonlinear ARDL models to reflect the long-term and short-term effects and asymmetry of oil price shocks and other factors on inflation expectations, we need to satisfy the variables' stationarity at I(0) or I(1) or a mixture of the two.
To examine the order of integration among the underlying variables in the absence of structural breakpoints, we applied the augmented Dickey-Fuller test (ADF) of Dickey and Fuller (1979), the Phillips-Perron test (PP) of Phillips and Perron (1988) and Kwiatkowski-Phillips-Schmidt-Shin test (KPSS) proposed by Kwiatkowski et al. (1992).As the test results shown in Tables 2 and 3, all the quarterly variables in China and the United States are stationary at the I(1) level and none of the variables are stationary at the I(2) level.Therefore, inflation expectations, oil price shocks, inflation rates, GDP, and EPU indices are all non-stationary data, and we can only analyze the regression relationship of non-stationary variables when these variables are guaranteed to be cointegrated.
However, this paper focuses on the impact of the epidemic as an important structural breakpoint, thus confirming that the impact of oil price shocks on inflation expectations is different before and after the epidemic, and the above unit root tests are not qualified to capture any structural breaks in the data.In order to avoid the potential misleading caused by this limitation, this paper introduces both the Perron unit root test (Perron, 1997) and the ZA test (Zivot & Andrew, 2002) to perform a unit root test with structural breaks on the quarterly data of variables in the United States and China.The results in Tables 4 and 5 show that, despite some variables being even stationary at I(0), the results of the tests with and without structural breaks resemblance.
Given that all variables satisfy the stationarity requirement, these results apply to both the ARDL model and Nonlinear ARDL model, meaning that both models can conveniently capture sequences with mixed integration orders.

Cointegration Test
The results of the unit root test show that almost all the underlying variables are stationary at I(1), that is, the horizontal series data is non-stationary.To further confirm the regression relationship, we need to test the cointegration between variables.According to the boundary test method in the ARDL-ECM, the null hypothesis of no cointegration in Eq.2 is  0 :  1 =  2 =  3 =  4 =  5 =  6 =  7 = 0.By comparing the critical value of the F statistic given by Pesaran et al7 .with the actual test results, we can choose whether to reject the null hypothesis.The specific principle is that if the actual F value is lower than the lower limit, it means that the null hypothesis is accepted, that is, there is no cointegration relationship between the variables; and when the actual F value exceeds the upper limit, we will reject the null hypothesis, indicating that there is a cointegration relationship between variables; if the F value is between the lower and upper values, it indicates that there is insufficient evidence to draw a conclusion.The cointegration test in Table 6 shows that the US data has an F-statistic of 10.804 > I(1) critical value = 4.43, which is significant at the 1% level, and similarly, China data has an F-statistic of 31.336> I(1) critical value = 4.43, significant at the 1% level.Therefore, the co-integration test results both help us reject the null hypothesis, concluding that there is a co-integration and long-run relationship between the variables in both countries.
Based on the above tests of the stationarity and cointegration of variables in the United States and China, the ARDL-ECM and Nonlinear ARDL model will further help us analyze the impact of multiple factors on inflation expectations before and after the COVID19 epidemic.

U.S.
To reflect the effect of COVID-19 outbreak on the oil price shocks in limited data, we introduce a dummy variable, which means that we consider the difference in the impact of oil price shocks on inflation expectations between the two sub-periods.Taking into account the specific time when the United States was affected by the epidemic, we define the structural breakpoints as the second quarter in 2020 and the third month in 2020 in quarterly and monthly samples, that is, the quarterly sample in the United States contains two sub-periods (2010Q1 to 2020Q1 and Q2 2020Q2 to 2021Q4), the monthly sample also includes two sub-periods (January 2010 to February 2020 and March 2020 to December 2021).

ARDL-ECM Results
According to the long-term cointegration between variables in the United States and the optimal lag option of AIC and SIC, we obtained the regression results of the American ARDL error correction model.Table 7 shows the results for the US benchmark model.
Given that oil price dynamics have an important impact on inflation, and inflation expectations have an anchoring effect on inflation, scholars have also devoted more enthusiasm to studying the relationship between oil price dynamics and inflation expectations.As a result, more research has emerged on the transmission mechanism of the impact of oil price changes on inflation expectations.Badel and McGillicuddy (2015) propose that with a tighter synchronization of all sources of oil price movements and inflation expectations, the correlation of American breakeven inflation expectations with oil prices in 2008-2015 is higher.
As shown in the table above, the overall quarterly data from 2010 to 2021 shows that the oil price shock has a significant impact on American households' inflation expectations in the short and long term.It has increased inflation expectations by 6% in the short-term and 7% in the long term, which is closely related to the consumption structure of American households.The main energy source of electricity consumption and transportation is oil so the rise in oil prices reduces the disposable income of American households, especially low-income families with more pressure during the epidemic, which also directly increases households' inflation expectations.Hammoudeh and Reboredo (2018) explore the non-linear impact of oil prices on US inflation expectations in different periods, arguing that the impact in the medium term is more significant than in the long term.After the outbreak, the level of U.S. households' inflation expectations increased by 5.6% in the short term and 6.5% in the long term due to insufficient output and disrupted logistics during the epidemic.The impact of oil price shocks on inflation expectations is also more significant after the epidemic.Compared with 2010-2019, oil price shock causes inflation expectations to increase by 2.8% and 3.2% in the short and long term after COVID-19 outbreak respectively.
With the outbreak of the COVID-19 epidemic in 2020, the world economies have been affected to varying degrees, and changes in inflation expectations also vary according to national conditions.Based on a survey of German citizens' inflation expectations since 2019, Coleman and Nautz (2020) propose that household inflation expectations during the epidemic are higher than the actual average inflation rate in Germany in the past.Rusiadi (2020) finds from the data of emerging market countries in 2019-2020 that the epidemic affects adaptive inflation expectations, and residents in emerging market countries have higher inflation expectations and lower purchasing power.Kapoor (2020) points out that the epidemic is with a rise in inflation expectations in India, followed by negative growth.As one of the most important energy sources for economic production and daily life, the historic rise in the oil price will further delay economic recovery and increase residents' living pressures and inflation expectations.Therefore, in the short term, the oil price shock has a positive impact on U.S. inflation expectations, and this impact has increased after the epidemic.By monitoring U.S. inflation expectations, Apergis and Apergis (2021) find that inflation expectations and their volatility are positively affected by the Covid-19 pandemic, which may signal a risk of inflation expectations breaking out of their anchors.Sharif (2020) analyzes the relationship between the spread of COVID-19 in the United States and the shock of oil price volatility in a time-frequency framework, explaining that geopolitical risks and economic policy uncertainty make the two more closely related.
By studying the relationship between the rise in oil prices during the epidemic and American households' inflation, Lee (2021) finds that the rise in oil prices will increase inflation expectations by increasing wage and the price of goods.In this paper, the lagged inflation expectations  −1 and  −2 have positive and negative effects on inflation expectations, respectively, where the impact of  −1 is more significant.A 1% increase in lagged inflation expectations will lead to a 44.7% increase in inflation expectations.This is the preliminary evidence that adaptive expectations and inflation expectations are adjusted due to past deviations, and also the persistent impact of price levels on current household inflation expectations.Inflation expectations are also affected by inflation rates and EPU over the same period, but as Cavallo, et, al. (2017) confirm, information frictions play a central role in the formation of household inflation expectations, with the lag of this friction, the impact is not significant.By contrast, the effect of GDP is more pronounced, with its magnitude and importance varying by different lag periods.The growth of GDP in the same period will effectively balance the price level, thereby reducing inflation expectations, and the continuous increase of GDP increases the confidence of residents, thereby further reducing their inflation expectations.
The diagnostic test in panel C shows that there is no autocorrelation problem between the variables.The BPG test shows heteroskedasticity in the data.However, given that we use the White test in the informatrix test to show no heteroscedasticity existence, and the method explains the consistent standard errors and covariances, heteroscedasticity is not a major issue from a statistical adequacy point of view.The Ramsey REST test indicated that the stability of the model could not be demonstrated at the 5% statistical level of significance, so to further test the stability of the estimates, we also performed CUSUM and CUSUMSQ, the results are shown in Figure 1 (see Appendix B).The CUSUM and CUSUMSQ parametric stability tests of inflation expectations show that the overall sample from 2010-2021 both remains within the 5% significance bound and therefore, the parameter estimates are stable.

Nonlinear ARDL Results
Combined with the analysis of the ARDL model, we know that the oil price shock increases the inflation expectation, especially after the outbreak of COVID19 outbreak.Moreover, we will use the Nonlinear ARDL model to explore the significance of the asymmetry of increasing and decreasing oil prices on inflation expectations, and the difference in the asymmetries of positive and negative oil price shocks before and after the epidemic.In order to achieve linear and nonlinear ARDL research for small samples, we improve the model by introducing dummy variables and interactive variables, to ensure the sample size and compare the differences before and after the structural break.This article mainly uses ARDL and Nonlinear ARDL models to estimate the asymmetric impact of oil price shocks on inflation expectations before and after the epidemic.The results are shown in Table 8: Consistent with the results of the ARDL error correction model, in the short term, both positive and negative shocks to oil prices have a significant positive impact on inflation expectations, which is in line with the assumption of asymmetry in oil price shocks, that is, the effect of the increase on the oil price is greater than that of negative oil prices.As we can see, the positive oil price shock causes inflation expectations to rise by 6.7% in the short term, and even the negative oil price shock brings 1.9% increase.In particular, the increase in oil prices over the same period will directly lead to a 3.2% increase in inflation expectations, while the decline in oil prices will only reduce inflation expectations by 0.8%.In the long run, this asymmetry is also obvious.Though the impact of positive oil price shocks is not significant, it is still difficult for negative oil price shocks to reverse the rising trend of inflation expectations.It should be pointed out that oil price increases with longer lags do not have a significant effect on inflation expectations, although they reduce inflation expectations.In the nonlinear model, whether in the long-term or the short-term, the effect of the dummy variable and the interactive variable is significant, indicating that the oil price shock has a positive impact on inflation expectations, which is strengthened during the epidemic.
The effect of the inflation rate and EPU on inflation expectations is still present but not significant.Short-term and long-term GDP growth can still reduce consumer inflation expectations, and the lagged GDP will affect the decline in residents' inflation expectations.For example, the lagged  −1 can even reduce residents' inflation expectations by 53.5%.The difference with the ARDL result is that in the short term, the lag period of inflation expectations  −1 has a significant negative impact on inflation expectations, but this still reflects the adjustment of adaptive expectations and inflation expectations due to past deviations.
Panel C demonstrates that the variables do not have problems with autocorrelation or heteroskedasticity, and the Ramsey REST test confirms that the model passes the standard test.More importantly, we examine the cointegration of the NARDL model and the asymmetry of oil price shocks in the short and long term, concluding that we can reject the null hypothesis of cointegration at the 1% significance level.Therefore, the positive impact and asymmetry of the 2010-2020 US oil price shock on inflation expectations can be confirmed by ARDL and Nonlinear ARDL models.

China
Similar to the data processing method in the United States, we consider the overall sample and introduce a dummy variable of structural disruption to compare the effects of oil price shocks before and after the epidemic.Due to the early start of the epidemic and the economic damage in China, we have the first quarter in 2020 and the second month in 2020 to split the quarterly and monthly samples respectively, that is, the quarterly sample in China contains two sub-periods (2010Q1-2019Q4 and 2020Q1-2021Q4), the monthly sample also includes two sub-periods (January 2010 to January 2020 and February 2020 to December 2021).

ARDL-ECM Results
According to the long-term cointegration relationship between variables in China and the optimal lags of AIC and SIC, we obtained the ARDL error correction model results of China's quarterly benchmark model, which are shown in Table 9 as follows.
As shown in Table 9, the overall quarterly data for 2010-2021 shows that China, like the US, is also significantly positively affected by the oil price shock on households' inflation expectations in the short and long term, the positive impact is less though.A 1% increase in oil prices will increase Chinese inflation expectations by 1.1% in the short term, and about 0.4% in the long run.This can also be explained by the consumption structure and consumption concept of Chinese residents, and China's isolation policy for COVID-19.In 2021, the per capita electricity consumption in China was 4989KWh compared to 12220KWh in the United States.Chinese households' electricity consumption grew by 5.9% year-on-year in 2020, while the consumption of gasoline used for travel has fallen by 3.53% year-on-year.Therefore, the impact of oil price shock on residents' disposable income is positive but limited.Yu (2022) studies the time-frequency dynamics of spillover effects between oil price shocks and global economic performance, and proposes that the recent outbreak of COVID-19 indicates that oil prices fluctuate significantly during the crisis, and that the impact of the epidemic on oil prices could even cause a serious impact on economic activities.Rafiuddin (2021) uses GCC member country data to show that although there is not much correlation in the short term, the impact of the global pandemic crisis on oil price shocks is significant in the medium and long term.
The above conclusions are also verified by the significant negative effect of the dummy variable.After the outbreak of the epidemic, the level of Chinese residents' inflation expectations decreased by 6.6% in the short term and 2.5% in the long term, and the negative impact of oil price shocks on inflation expectations was also more significant after the epidemic.Compared with 2010-2019, the impact of oil price shocks after COVID-19 outbreak on inflation expectations was reduced by 8.3% and 3.1% in the short and long term, respectively.This fully reflects that Chinese residents' reliance on oil prices has decreased significantly based on the less use of transportations according to the isolation policy.Therefore, with the decreasing consumption of oil products, the impact of oil prices on household inflation expectations has also declined.
It can be seen that the Chinese inflation expectations are more dependent on the lagged inflation expectations  −1 and  −2 .Interestingly, long-duration lagged inflation expectations have a significant negative impact on current inflation expectations.Part of the reason is that according to the Chinese government's macroeconomic regulation policies, Chinese residents tend to trust the government to maintain its stability after a prolonged period of inflation.Another part of the reason is that China's inflation expectations also have adaptive expectations and inflation expectations are adjusted due to past deviations.According to the results, inflation expectations are also significantly affected by the lagged inflation rate  −1 , the lagged  −2 , and the lagged  −1 .The positive effects of the inflation rate and EPU and the negative effects of GDP are in line with the law of economic development.
The diagnostic tests in panel C show that there are no problems with autocorrelation and heteroskedasticity between variables.The Ramsey REST test showed that the model did not pass the standard test at the 5% statistical level of significance, so to further test the stability of the estimates, we also performed CUSUM and CUSUMSQ, the results are shown in Figure 2 (see Appendix B).The CUSUM and CUSUMSQ parametric stability tests of inflation expectations show that the overall sample in 2010-2021 is briefly out of bounds, but the subject remains within the 5% significance bound, so the parameter estimates can be considered stable.

Nonlinear ARDL Results
Nasir et al. (2020) apply the Nonlinear ARDL model to the assessment of the relationship between oil price dynamics and inflation expectations in New Zealand and the United Kingdom, and found that changes in oil prices have asymmetric effects on inflation expectations, while the exchange rate, money supply, Output growth, unemployment, and fiscal deficit/surplus also play a role in this pass through.Similarly, based on the Nonlinear ARDL model, we further explore the asymmetry of positive and negative oil price shocks and the change in the impact of oil price shocks after the epidemic.The results are shown in Table 10: In the Nonlinear ARDL model, which explores the asymmetry of oil price shocks, we reach a conclusion similar to that of the United States, that is, in the short term, the impact of positive and negative oil prices on inflation expectations is asymmetric, and the positive oil price change plays a bigger role on inflation expectations than the negative change.As we can see from the table, a positive oil price shock causes inflation expectations to rise by 3.7% in the short term, while a negative oil price shock also gives a positive impact which is only 1.0% though.In particular, the increase in oil prices over the same period directly increases the inflation expectation by 2.4%, and the lagged decline in oil prices also significantly increases inflation expectations by up to 1.1%.In the long run, this asymmetry also exists.Although the rate of increase is not obvious, the shock of negative oil prices is still working on the increase of inflation expectations.In the short term, the dummy variable and the interactive variables are also important in China.After the COVID-19 outbreak, the oil price shock harms inflation expectations, the impact is not significant in the long term though.
The inflation rate and EPU have a very significant positive impact on China's inflation expectations both in the short and long term, indicating that the rise in Chinese residents' inflation expectations is affected by the pass-through of inflation and economic policy uncertainty during the epidemic.However, oil prices not only do not lead to an increase in expectations, but even had the opposite effect due to the reduction in consumer demand.With all the significant effects GDP brings inflation expectation are positive.Changes in GDP with different lag periods have different effects on inflation expectations.For example, the lagged  −3 even reduces households' inflation expectations by 64.9%.
The diagnostic tests in panel C demonstrate that the variables do not have problems with autocorrelation or heteroskedasticity, and the Ramsey REST test confirms that the model passes the standard test.According to the boundary and asymmetry tests, we can reject the cointegration null hypothesis at a significant level of 1% and reject the null hypothesis of symmetry at the 1% significance level.The negative impact and asymmetry of China's oil price shock on inflation expectations from 2010 to 2020 can be confirmed by ARDL and Nonlinear ARDL models.
Last but not least, the Nonlinear ARDL model has a better fit for both the US and China quarterly data than the ARDL model, which further shows that the asymmetric effect of oil price shocks on inflation expectations is more in line with the current state of economic development in the U.S. and China.

Robustness Check
In this section, we will apply monthly data from the US and China for robustness checks.As described in the previous section, we expanded the US and China data into the overall sample and two subsamples, the subsamples are January 2010-February 2020 and March 2020-December 2021 in the US, and January 2010-January 2020 and February 2020-December 2021 in China.Because of the supplementation of data and the setting of subsamples, we no longer need to use dummy variables and interactive variables but draw conclusions by directly comparing the results of the two subsamples.
Before this, we also perform unit root tests and boundary tests for the monthly data, and the results are shown in Figures A1-A5 (in Appendix A).The results also prove that all variables are stationary at I(0) or I(1), and there is cointegration between variables.

Causality Test
We cannot conclude the causal relationship between oil price shocks and inflation expectations although we have explored the linear impact and nonlinear symmetry of oil price shocks in the benchmark model.Therefore, we use the Toda-Yamamoto Granger causality test proposed by Toda and Yamamoto (1995).Compared with the classical Granger test, the advantages of the Toda and Yamamoto process are given as follows: First, the Granger test can give spurious regressions on aggregate variables for functions with time lags.Second, the F statistic can only be used when the variables are cointegrated.Third, the Toda and Yamamoto tests for Granger non-causality are based on the modified Wald test (MWald) and the seemingly uncorrelated regression model (SUR model).Thus, the Toda and Yamamoto procedure minimizes the risk of determining the optimal lag order for each variable, and it works for all variables with or without stationarity and cointegration.
However, the Toda-Yamamoto test cannot capture the effect of the presence of structural breakpoints on causality.
To make up for this deficiency, we conduct this test also in the subsamples.The specific test results are shown in Table 11.
The null hypothesis of the test is that there is no causality between the oil price shock and inflation expectations.When the p-value is less than 0.5, we have reason to reject the null hypothesis.As shown in the results of the table above, only in the COVID-19 regime, did the oil price shock causes inflation expectations.While the oil price shock is the cause of inflation expectations in all the Chinese samples.On the contrary, except for the COVID-19 regime, other samples does not reflect that inflation expectations can cause oil price change.This exception may be due to that the impact of the pandemic and inflation expectations has reduced demand for oil, thereby affecting its price.

U.S.
Table 12 is the ARDL error correction model results.
Consistent with the conclusion of the benchmark ARDL-ECM, the full sample gives us positive and significant effects from the oil price shock in the short term, with an impact level of 8.0%.The effects of lagged inflation expectations and inflation on inflation expectations are similar to the benchmark results.The difference is that, in the long run, the oil price shock is no longer the factor that significantly affects inflation expectations, but the inflation rate is.
In the two subsamples, we can also see that the impacts of oil price shocks on inflation expectations are equally positive and significant in the short and long term.The impact of oil price shocks on inflation expectations under the COVID-19 regime has increased significantly, which is from 0.3% to 2.5% in the short term, from 0.7% to 1.9% in the long run, which also shows the importance of the COVID19 outbreak as a structural breakpoint.
Besides, the oil price shock has a more significant effect on inflation expectations in the long run under the pre-COVID19 regime, and during the epidemic, inflation expectations are also significantly affected by inflation and GDP.
Table 13 shows a robustness test for the asymmetry of positive and negative oil price shocks using the Nonlinear ARDL model.
In the short-term and long-term of the overall sample, both positive and negative oil prices have a significant positive impact on inflation expectations, and the magnitude of positive shocks is greater than that of negative shocks, which more intuitively proves price stickiness, and also shows the asymmetry of oil prices.Interestingly, in the results of the two subsamples, the positive effect of negative oil price shocks on inflation expectations even exceeds that of positive oil price shocks, which further reflects the significantly positive effect of oil price shocks and the more obvious price stickiness.The asymmetry is further amplified in the monthly data.
Similarly, we also conduct the robustness test on the Chinese data.See Tables 14 and 15.
According to the overall sample in Table 14, oil price shocks have a negative but insignificant impact on inflation expectations in the short and long term.The lagged inflation expectations exert a persistent and significant positive impact, although the short-term impact varies with the choice of lag period.In the two subsamples, the oil price shock has a positive impact on inflation expectations in the long run, while the rise in oil prices under the epidemic regime has a significant negative impact on inflation expectations, which is consistent with the conclusion of the benchmark model.
It is worth noting in Table 15 that positive oil prices in the full sample negatively impact inflation expectations in the short term, while negative oil prices increase inflation expectations.This can be explained by the sub-sample.
Under the pre-COVID19 regime, both positive and negative oil prices have a positive impact on inflation expectation, and the impact of negative oil prices is more significant, which fully reflects the asymmetry of positive and negative oil price shocks, that is, the impact of positive oil prices on inflation expectations is relatively greater than that of negative oil price.Under the COVID19 mechanism, oil price shock has a significant negative impact on China's inflation expectations, which once again reflects the asymmetry of positive and negative oil price shocks, but in the overall sample, the role of negative impact is even more significant because of the outbreak, which also explains the long-term results of the overall sample that both positive and negative oil price shocks have a negative, although insignificant, impact on inflation expectations.
Based on the above robustness check, we have once again verified that the impact of oil prices on inflation expectations has changed after COVID-19 outbreak.At the same time, it is confirmed that the asymmetry of positive and negative oil price shocks exists in the United States and China.However, the positive effect of the positive oil price shock after the epidemic is strengthened in the United States, and the negative effect of China's negative oil price shock is more obvious.

Further Discussion
The asymmetric impact of crude oil prices on the households' inflation expectations is essentially a shock of energy prices.Changes in crude oil prices act more directly on the energy sector and the prices of refined products, which in turn raise residents' expectations of future price levels by increasing the prices of oil products.Ultimately, crude oil prices have an impact on the inflation expectations of the population.More generally, increases in crude oil prices have a more significant impact on inflation expectations by generating price volatility in oil products than the impact of decreases.For example, gasoline prices have risen sharply and swiftly following a rise in crude oil prices-such as occurred in 1999 and 2000 and during the Gulf War in 1990 in America.This asymmetric relationship between crude oil and product prices within the energy sector may also have an asymmetric impact on other macroeconomic variables.
The impact of oil price change is important, where both supply-and demand-level shocks may be responsible for its asymmetric effects.Oil, as an essential basic energy source and industrial raw material, directly affects the functioning of the economy and the consumption structure and quality of residents' lives.The impact of supply-side shocks caused by oil price changes on economic variables is not as significant in China as the impact of demand shocks.However, China, as the world's largest oil importer, is very dependent on oil.Supply shocks from higher oil prices can directly affect GDP growth and the timely replenishment of market output, which in turn affects price levels and residents' inflation expectations.In terms of demand shocks, oil price fluctuations from demand shocks can boost the economy and, conversely, increase inflation and further raise residents' inflation expectations.The impact of falling oil prices is more in terms of raising demand and boosting economic development, which in turn raises residents' inflation expectations.Thus, there is an asymmetry in the change of inflation expectations caused by the rise and fall of oil prices.
The impact of positive and negative changes in oil prices on economic uncertainty is asymmetric.The increase in oil prices leads to an increase in economic uncertainty, which affects the development of macroeconomics and the stability of financial markets.These fluctuations can affect the sustainability and stability of the population's consumption and investments and may also cause the energy market to raise preventive energy reserves and speculative activity in financial markets, which can lead to higher production costs and higher price levels.
Inflation expectations are more likely to increase in an unstable economic environment and thus react asymmetrically to increases and decreases in oil prices.

Conclusion
The primary objective of this study is to figure out the asymmetry effects of oil price changes on U.S. and China household inflation expectations after COVID-19 outbreak.By adopting the ARDL model and the Nonlinear ARDL model to explore the dynamic relationship between oil price shocks and household inflation expectations, we analyze the data from the first quarter of 2010 to the fourth quarter of 2021.To verify the applicability of the models, we confirm the stationarity and cointegration relationships in the short and long term by unit root tests and boundary tests, and also apply diagnostic tests to demonstrate the absence of autocorrelation, heteroscedasticity, and stability issues, thereby increasing the credibility of the conclusions.In the robustness test, we use the Toda-Yamamoto causality test to reflect the causal relationship between the oil price shock and inflation expectations from January 2010 to December 2021, and further through the comparison of sub-samples in different regimes to check the reliability of the conclusions of the benchmark model as follows.
1.The outbreak of COVID-19 has changed the impact of oil price dynamics on household inflation expectations.
The positive impact of oil prices in the U.S. has been amplified, while diminished in China.
2. There are asymmetric effects from positive and negative oil price changes.Consistently, before the COVID-19 outbreak, inflation expectations in both the U.S. and China were more affected by positive oil price shocks than negative oil price shocks, so there is an interesting phenomenon that negative oil price shocks raised inflation expectations, while in the COVID-19 regime, the asymmetry persists despite changes between positive and negative shocks.t 3. Finally, the asymmetry of the oil price shock between the United States and China has diverged in the wake of the pandemic.The role of the positive oil price shock in the United States has been further strengthened after the epidemic, while in China, on the contrary, the role of the negative oil price shock is more significant after the outbreak, so there is an interesting phenomenon that the positive oil price shock reduces household inflation expectations.
According to the empirical analysis, we can see the similarities in the economic development of the United States and China, but also recognize the differences in economic development between the two countries due to the impact of the epidemic.On the one hand, this is related to differences in the epidemic isolation policies in the two countries, and on the other hand, it is also related to the households' consumption structure of the two countries.Under the current situation of rising oil prices and the urgent need for economic recovery, how to optimize the production structure and stabilize residents' inflation expectations is a problem that both countries need to solve, and it is also of great significance to the sustainable development of the two countries' economies.
Pesaran, Shin, and Smith (2001)rator;   captures the error term.1, 2,…, 7 are the optimal lag orders of different variables selected by Akaike information criterion (AIC) and Schwarz information criterion (SIC).For the coefficients,  1 ,  2 ,…,  7 represent the short run while  1 ,  2 ,…,  7 represent the long run.After obtaining the ARDL model, we need to use the unit root test to determine whether the included dependent variables are stationary at the level sequence I(0) or the first-order difference I(1).When all variables are stationary, we can confirm cointegration between variables using boundary tests based on F and t correlation statistics(McNown et al. 2018).Pesaran, Shin, and Smith (2001)generated key table values or F and t-statistics obtained from the analysis, and in this paper, we use only F statistics 5 .The F-statistic asymptotic table critical values can be used with two different bounds, where the variable is assumed to be stationary at either I(0) or I(1).If the calculated F-statistic exceeds the critical values, a clear conclusion can be drawn.However, if the calculated value is between the two bounds, it is impossible to draw firm conclusions about the cointegration relationship without knowing the stability of the variables.Long run coefficients are  1 ,  2 ,  3 ,  4 ,  5 ,  6 ,  7 ;   is the constant term.The long-run coefficient of oil price changes pass-through to inflation expectation can be defined as   = − .To check our model's reliability, we employed several diagnostic tests, such as autocorrelation, normality distributed, and heteroskedasticity tests.Besides, we also used the Cumulative Sum of Recursive Residuals test (CUSUM test) and Cumulative Sum of Square Recursive Residuals test (CUSUMSQ test) to test the stability of the short-run and the long-run coefficients.
China and the United States.As important economies in the world, China and the United States are important importers of crude oil.As the second largest importer of oil and crude oil in the world, the U.S. import of oil was 12 MMb/d 6 in 2010 and 9.1 MMb/d in 2019, of which 9.2 MMb/d and 6.8 MMb/d of which were crude oil.Since 2001, China's crude oil imports have been rising for 20 consecutive years.In 2017, China's crude oil imports reached 8.4 million barrels per day, making it the country with the largest total crude oil imports.The COVID-19 outbreak had a huge impact on China and the United States, not only in oil imports but also in oil prices, GDP, inflation, and EPU index.The United States imported about 7.86 MMb/d of petroleum in 2020, which included 5.88 MMb/d of crude oil and 1.98 MMb/d of non-crude petroleum liquids and refined petroleum products.These were the lowest levels of imports of total petroleum and crude petroleum products.oil since 1991.According to data released by the National Bureau of Statistics of China, China's crude oil imports in 2021